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PRINCIPLES 

OF 

TRANSFORMER  DESIGN 


PRINCIPLES 


OF 


TRANSFORMER  DESIGN 


BY 

ALFRED   STILL 

M.INST.C.E.,    FEL.A.I.E.E.,   M.I.E.E. 

Professor  of  Electrical  Engineering,  Purdue  University, 

Author  of  "Polyphase  Currents," 

"Electric  Power  Transmission,"  etc. 


NEW  YORK: 

JOHN  WILEY  &  SONS,  Inc. 

London:  CHAPMAN  &  HALL,  Limited 


fi.  C.  StoU  CoUcge 


Copyright,  1919,  by 
ALFRED   STILL 


PRESS    OF 
BRAUNWORTH  k  CO. 
10/24  BOOK  MANUFACTURERS 

BROOKLVN,   N.   V. 


PREFACE 


A  BOOK  which  deals  exclusively  with  the  theory 
and  design  of  alternating  current  transformers  is  not 
likely  to  meet  the  requirements  of  a  College  text  to 
the  same  extent  as  if  its  scope  were  broadened  to  include 
other  types  of  electrical  machinery.  On  the  other 
hand,  the  fact  that  there  may  be  a  Hmited  demand  for 
it  by  college  students  taking  advanced  courses  in  elec- 
trical engineering  has  led  the  writer  to  follow  the 
method  of  presentation  which  he  has  found  successful 
in  teaching  electrical  design  to  senior  students  in  the 
school  of  Electrical  Engineering  at  Purdue  University. 
Stress  is  laid  on  the  fundamental  principles  of  electrical 
engineering,  and  an  attempt  is  made  to  explain  the 
reasons  underlying  all  statements  and  formulas,  even 
when  this  involves  the  introduction  of  additional 
material  which  might  be  omitted  if  the  needs  of  the 
practical  designer  were  alone  to  be  considered. 

A  large  portion  of  Chapter  II  has  already  appeared 
in  the  form  of  articles  contributed  by  the  writer  to  the 
Electrical  World;  but  the  greater  part  of  the  material 
in  this  book  has  not  previously  appeared  in  print. 

Lafayette,  Ind.  T  K^ 66  1  22854 

January,  1919  ^  <-^ 


CONTENTS 


PAGB 

Preface iii 

List  of  Symbols ix 


CHAPTER  I 

Elementary  Theory. — Types.— Construction 

ART, 

1.  Introductory i 

2.  Elementary  Theory  of  Transformer 2 

3.  Effect  of  Closing  the  Secondary  Circuit 6 

4.  Vector  Diagrams  of  Loaded  Transformer  without  Leakage 10 

5.  Polyphase  Transformers 12 

6.  Problems  of  Design 13 

7.  Classification  of  Alternating-current  Transformers 14 

8.  Tj^es  of  Transformers. — Construction 17 

9.  Mechanical  Stresses  in  Transformers 24 

CHAPTER  II 

Insulation  of  High-pressure  Transformers 

10.  The  Dielectric  Circuit 32 

11.  Capacity  of  Plate  Condenser 40 

12.  Capacities  in  Series 42 

13.  Surface  Leakage 46 

14.  Practical  Rules  Applicable  to  the  Insulation  of  High-voltage 

Transformers 48 

15.  Winding  Space  Factor 51 

16.  Oil  insulation 52 

17.  Terminals  and  Bushings 54 

18.  Oil-filled  Bushing 57 

19.  Condenser-type  Bushing 62 

V 


VI  CONTENTS 

CHAPTER  III 
Efficiency  and  Heating  of  Transformers 

PAGE 

20.  Losses  in  Core  and  Windings 69 

21.  Efficiency 73 

22.  Temperature  of  Transformer  Windings 79 

23.  Heat  Conductivity  of  Insulating  Materials 80 

24.  Cooling  Transformers  by  Air  Blast 88 

25.  Oil-immersed  Transformers,  Self-cooling 91 

26.  Effect  of  Corrugations  in  Vertical  Sides  of  Containing  Tank ...  94 

27.  Effect  of  Overloads  on  Transformer  Temperatures 98 

28.  Self-cooling  Transformers  for  Large  Outputs 103 

29.  Water-cooled  Transformers 105 

30.  Transformers  Cooled  by  Forced  Oil  Circulation 106 


CHAPTER  IV 
Magnetic  Leakage  in  Transformers.— Reactance.— Regulation 

31.  Magnetic  Leakage 107 

32.  Effect  of  Magnetic  Leakage  on  Voltage  Regulation 109 

33.  Experimental  Determination  of  the  Leakage  Reactance  of  a 

Transformer 114 

34.  Calculation  of  Reactive  Voltage  Drop 117 

35.  Calculation  of  Exciting  Current 125 

36.  Vector   Diagram    Showing   Effect   of   Magnetic   Leakage   on 

Voltage  Regulation  of  Transformers 132 


CHAPTER  V 

Procedure  in  Transformer  Design 

37.  The  Output  Equation 138 

38.  Specifications 140 

39.  Estimate  of  Number  of  Turns  in  Windings 141 

40.  Procedure  to  Determine  Dimensions  of  a  New  Design 149 

41.  Space  Factors 151 

42.  Weight  and  Cost  of  Transformers 151 

43.  Numerical  Example 154 


CONTENTS  Vii 

CHAPTER  VI 

Transformers  for  Special  Purposes 

PAGE 

44.  General  Remarks 177 

45.  Transformers  for  Large  Currents  and  Low  Voltages 177 

46.  Constant  Current  Transformers 1 78 

47.  Current  Transformers  for  use  with  Measuring  Instruments 183 

48.  Auto-transformers 191 

49.  Induction  Regulators 197 


LIST  OF  SYMBOLS 


i4=area  of  equipotential  surface  perpendicular  to  lines  of  force 

(sq.  cm.). 
A  =  cross-section  of  iron  in  plane  perpendicular  to  laminations  (sq.  in.), 
a = ampere-turns  per  inch  length  of  magnetic  path, 
a = total  thickness  of  copper  per  inch  of  coil  measured  perpendicularly 

to  layers. 

5  =  magnetic  flux  per  sq.  cm.  (gauss). 
Bam    is  defined  in  Art.  9. 

b  =  total  thickness  of  copper  per  inch  of  coil  measured  through  insu- 
lation parallel  with  layers. 

C= electrostatic  capacity;  or  permittance, 

coulombs      -  .  c    re       J\ 

= r- —  =nux  per  unit  e.m.i.  (farad). 

Cm/=  capacity  in  microfarads. 

c=a  coefficient  used  in  determining  Vt. 

D = flux  density  in  electrostatic  field = — = KkG  (coulombs  per  sq.  cm.). 

£=e.m.f.  (volts),  usually  r.m.s.  value,  but  sometimes  used  for  max. 

value. 
£1  =  virtual  value  of  induced  volts  in  primary  (  =£2X-^) . 

£'1  =  component  of  impressed  voltage  to  balance  Ei. 
£2  =  secondary  e.m.f.  produced  by  flux  <t>;  induced  secondary  e.m.f. 
£e  =  primary    voltage    equivalent    to    secondary    terminal    voltage 


(-■xff)' 


X  LIST  OF  SYMBOLS 

Ep  =  e.m.f.  (volts)  applied  at  primar>'  terminals. 
£i  =  secondary  terminal  voltage. 

£z  =  impressed  primary  voltage  when  secondary  is  short-circuited. 
e=e.m.f.  (volts). 

F= force  (dynes). 

/=  frequency  (cycles  per  second). 

de 
G=3T  =  potential  gradient  (volts  per  centimeter). 

g  =  distance  between  copper  of  adjacent  primary  and  secondary 
coils,  in  centimeters  (Fig.  42). 

fl'  =  magnetizing  force,  or  m.m.f.  per  cm. 
A  =  length  (cms.)  defined  in  text  (Fig.  42). 

7  =  r.m.s.  value  of  current  (amps.). 
/i  =  balancing  component  of  primary  current  =Isljr]. 

7c  =  current  in  the  portion  of  an  auto-transformer  winding  common 

to  both  primary  and  secondary  circuits. 
7e  =  total  primary  exciting  current. 

7o  =  "  wattless"  component  of  h  (magnetizing^  component). 
Ip  =  total  primary  current. 
7,  =  total  secondary  current. 
/„,  =  "  energy  "  component  of  !,■  ("  in-phase  "  component). 

K  =  8.84X10-^*  farads  per  cm.  cube  =  the  specific  capacity  of  air. 

''  >  definition  follows  formula  (34)  in  Art.  27. 

Kv  =  kilovolts. 
jfe  =  dielectric  constant  or  relative  specific  capacity,  or  permittivity 

{k  =  i  for  air). 
Jfe=heat  conductivity  (watts  per  inch  cube  per  1°  C). 
jfe  =  coefficient  used  in  calculating  the  efifective  cooling  surface  of 

corrugated  tanks. 
jfec  =  about  1. 8X10-8  for  copper. 
ifei  =  (refer  text  (Art.  39)  for  definition). 

/  =  length  (cms.). 

/  =  mean  length,  in  centimeters,  of  projecting  end  of  transformer  coiL 

/  =  length  measured  along  line  or  tube  of  induction  (cms.).' 


LIST  OF  SYMBOLS  xi 

/c=mean  length  per  turn  of  windmgs. 

/i=mean  length  of  magnetic  circuit  measured  along  flux  lines. 

Mc  =  weight  of  copper  in  transformer  coils  (lbs.). 
Afo  =  weight  of  oil  in  transformer  tank  (lbs.). 

M  =  —  (in  formula  for  calculating  cooling  surface  of  corrugated  tanks). 

M  =  usually  from  1.6  to  2  in  B"  (core  loss  formulas). 

P  =  weight  of  iron  in  transformer  core  (or  portion  of  core),  lbs. 

^= thickness  of  half  primary  coil  in  centimeters  (defined  in  text  in 
connection  with  Fig.  42). 

i?= resistance  (ohms). 
/?i  =  resistance  of  primary  winding  (ohms). 
J?2= resistance  of  secondary  winding  (ohms). 
Rh  =  "  thermal  ohms." 
i?p  =  equivalent  primary  resistance  =  i?i+i?2  ij?)  • 

total  number  of  turns  ,  ,  . 

r  =  ratio r 77 .    .    ^.     ■ r-  (auto- transformers). 

number  of  turns  common  to  both  circuits 

5  =  effective  cooling  surface  of  transformer  tank  (sq.  in). 

s  =  thickness  of  half  secondary  coil  (cms.)  defined  in  text  (Fig.  42). 

7'= number  of  turns  in  coil  of  wire. 
ri  =  number  of  turns  in  half    primary  group  of   coils  adjacent  to 

secondary  coil. 
Z'2= number  of  turns  in  half  secondary  group  of  coils  adjacent  to 

primary  coQ. 
7'd= difference  of  temperature  (degrees  centigrade). 
To = initial  oil  temperature. 
7'p= number  of  turns  in  primary  winding. 
Ts  =  number  of  turns  in  secondary  winding. 
Tt  =  o)l  temperature  at  end  of  time  im  minutes. 

/  =  thickness  (usually  inches). 

/  =  interval  of  time  (seconds). 
<7?i  =  interval  of  time  (minutes). 

7t= volts  induced  per  turn  of  transformer  winding. 


ai  LIST  OF  SYMBOLS 

W  =  power  (watts). 
IFc  =  full-load  copper  loss  (watts). 
Wi  =  core  loss  (watts). 
Wt  =  toU\  transformer  losses  (watts). 
tr  =  watts  dissipated  per  sq.  in.  of  (effective)  tank  surface. 
w  =  watts  lost  per  lb.  of  iron  in  (laminated)  core. 

Xi  =  reactance  (ohms)  of  one  high-low  section  of  winding. 
Xp  =  reactance  (ohms)  commonly  referred  to  as  equivalent  primaiy 
reactance. 

Zp  =  impedance  (ohms)  on  short  circuit. 

0  =  phase  angle  (cos  0  =  power  factor  of  external  circuit), 
ff  =  "  electrical "  angle  (radians)  =2irft. 
X  =  pitch  of  corrugations  on  tank  surface. 
*  =  magnetic  flux  (Ma.xwells)  in  iron  core. 

0  =  phase  angle  (cos  0= power  factor  on  primary  side  of  transformer). 
'^  =  dielectric  flux,  or  quantity  of  electricity,  or  electrostatic  induc- 
tion =  CE  =  AD  coulombs. 


PRINCIPLES 

OF 

TRANSFORMER   DESIGN 


CHAPTER  I 

ELEMENTARY  THEORY— TYPES— CONSTRUCTION 

1.  Introductory.  The  design  of  a  small  lighting 
transformer  for  use  on  circuits  up  to  2200  volts,  or 
even  6600  volts,  is  a  very  simple  matter.  The  items 
of  importance  to  the  designer  are: 

(i)  The  iron  and  copper  losses;  eflficiency,  and  tem- 
perature rise; 

(2)  The  voltage  regulation,  which  depends  mainly 
upon  the  magnetic  leakage,  and  therefore  upon  the 
arrangement  of  the  primary  and  secondary  coils; 

(3)  Economical  considerations,  including  manufac- 
turing cost. 

With  the  higher  voltages  and  larger  units,  not  only 
does  the  question  of  adequate  cooling  become  of  greater 
importance;  but  other  factors  are  introduced  which 
call  for  considerable  knowledge  and  skill  on  the  part 
of  the  designer.    The  problems  of  insulation  and  pro- 


PltOFERTY  UBRAK: 

N.  C.  Stau  Collm 


2  PRINCIPLES  OF  TRANSFORMER  DESIGN 

tection  against  abnormal  high-frequency  surges  in  the 
external  circuit  are  perhaps  the  most  important;  but 
with  the  increasing  amount  of  power  dealt  with  by 
some  modern  units,  the  mechanical  forces  exerted  by 
the  magnetic  flux  on  short-circuits,  or  heavy  over 
loads,  may  be  enormous,  requiring  special  means  of 
clamping  or  bracing  the  coils,  to  prevent  deformation 
and  damage  to  insulation. 

Since  we  are  concerned  mainly  with  a  study  of  the 
transformer  from  the  view  point  of  the  designer,  Uttle 
will  be  said  concerning  the  operation  of  transformers, 
or  the  advantages  and  disadvantages  of  the  different 
methods  of  connecting  the  units  on  polyphase  systems. 
It  will,  however,  be  necessary  to  discuss  the  theory 
underlying  the  action  of  all  static  transformers,  and 
it  is  proposed  to  take  up  the  various  aspects  of  the 
subject  in  the  following  order : 

Elementary  theory,  omitting  all  considerations  likely 
to  obscure  the  fundamental  principles;  brief  descrip- 
tion of  leading  types  and  methods  of  manufacture; 
problems  connected  with  insulation;  losses,  heating, 
and  efficiency;  advanced  theory,  including  study  of 
magnetic  leakage  and  voltage  regulation;  procedure 
in  design;  numerical  examples  of  design;  reference  to 
special  types  of  transformers. 

2.  Elementary  Theory  of  Transformer.  A  single- 
phase  alternating  current  transformer  consists  essen- 
tially of  a  core  of  laminated  iron  upon  which  are  wound 
two  distinct  sets  of  coils,  known  as  the  primary  and 
secondary  windings,  respectively,  all  as  shown  dia- 
grammatically  in  Fig.  i. 


ELEMENTARY  THEORY— TYPES— CONSTRUCTION       3 

When  an  alternating  e.m.f.  of  Ep  volts  is  applied 
to  the  terminals  of  the  primary  (P),  this  will  set  up  a 
certain  flux  {^)  of  alternating  magnetism  in  the  iron 
core,  and  this  flux  will,  in  turn,  induce  a  counter  e.m.f. 
of  self-induction  in  the  primary  winding;  the  action 
being  similar  to  what  occurs  in  any  highly  inductive 
coil  or  winding.     Moreover,  since  the  secondary  coils — 


^^ 

-r 

^s 

. 

m 

'              < 

■■■ 

1 

1 

Ep^ 

Olt8< 

Piy)< 

1 

(S'up 

WEllS 

ndingB.        y 

I^M 

' 

V. 

m 

• — ^ 

turns 

^Pathol  flu 
linking  wl 

< 
Taturr 

x:^  mar 
th  botli  wl 

E,  /oita 

(Liad) 


Fig.  I. — Essential  Parts  of  Single-phase  Transformer. 


although  not  in  electrical  connection  with  the  pri- 
mary— are  wound  on  the  same  iron  core,  the  variations 
of  magnetic  flux  which  induce  the  counter  e.m.f.  in 
the  primary  coils  will,  at  the  same  time,  generate  an 
e.m.f.  in  the  secondary  winding. 

The  path  of  the  magnetic  lines  is  usually  through 
a   closed    iron  circuit    of  low  reluctance,  in  order  that 


4  PRINCIPLES  OF  TRANSFORMER  DESIGN 

the  exciting  ampere-turns  shall  be  small.  There  will 
always  be  some  flux  set  up  by  the  primary  which  does 
not  Unk  with  the  secondary,  but  the  amount  of  this 
leakage  flux  is  usually  very  small,  and  in  any  €ase 
it  is  proposed  to  ignore  it  entirely  in  this  preliminary 
study.  In  this  connection  it  may  be  pointed  out  that 
the  design  mdicated  in  Fig.  i,  with  a  large  space  for 
leakage  flux  between  the  primary  and  secondary  coils, 
would  be  unsatisfactory  in  practice;  but  the  assump- 
tion will  now  be  made  that  the  whole  of  the  flux 
($  maxwells)  which  passes  thfough  the  primary  coils, 
links  also  with  all  the  secondary  coils.  In  other  words, 
the  e.m.f.  induced  in  the  winding  per  turn  oj  wire  will 
be  the  same  in  the  secondary  as  in  the  primary 
coils. 

Suppose,  in  the  first  place,  that  the  two  ends  of 
the  primary  winding  are  connected  to  constant  pres- 
sure mains,  and  that  no  current  is  taken  from  the 
secondary  terminals.  The  total  flux  of  $  maxwells 
increases  twice  from  zero  to  its  maximum  value,  and 
decreases  twice  from  its  maximum  to  zero  value,  in 
the  time  of  one  complete  period.  The  flux  cut  per 
second  is  therefore  4$/,  and  the  average  value  of  the 
induced  e.m.f.  in  the  primary  is, 

^4^/T. 

■'-'average  g      VOltS, 

where  Tp  stands  for  the  number  of  turns  in  the  primary 
winding. 

If  we  assume  the  flux  variations  to  be  sinusoidal,  the 


ELEMENTARY  THEORY— TYPES— CONSTRUCTION       5 

form  factor  is  i .  1 1 ,  and  the  virtual  value  of  the  induced 
primary  volts  will  be, 


E  =444/iZ^^ 


(i) 


The  vector  diagram  corresponding  to  these  condi- 
tions has  been  drawn  in  Fig.  2.  Here  OB  represents  the 
phase  of  the  flux  which  is  set  up  by  the  current  Oh  in 


/ 
/ 

E,' 

I.                                                    \ 

\ 
\ 
t 

3                                                           'E 

Fig.  2. — Vector  Diagram  of  Unloaded  Transformer. 

the  primary.  This  total  primary  exciting  current  can 
be  thought  of  as  consisting  of  two  components:  the 
"  wattless  "  component  Oh  which  is  the  true  magnetiz- 
ing current,  in  phase  with  the  flux;  and  01  w  (which  owes 
its  existence  to  hysteresis  and  eddy  current  losses) 
exactly  90°  in  advance  of  the  flux.  The  volts  induced 
in  the  primary  are  OEi  drawn  90°  behind  OB  to  repre- 
sent the  lag  of  a  quarter  period.  The  voltage  that  must 
be  impressed  at  the  terminals  of  the  primary  is  OEp 
made  up  of  the  component  OE'i  exactly  equal  but 
opposite  to  OEi,  and  E'lEp  drawn  parallel  to  0/«  and 


6  PRINCIPLES  OF  TRANSFORMER  DESIGN 

representing  the  IR  drop  in  the  primary  circuit.  The 
actual  magnitude  of  this  component  would  be  hRi 
where  Ri  is  the  ohmic  resistance  of  the  primary;  but  in 
practice  this  ohmic  drop  is  usually  so  small  as  to  be 
neghgible,  and  the  impressed  voltage  Ep  is  virtually  the 
same  as  E'l,  i.e.,  equal  in  amount,  but  opposite  in  phase 
to  the  induced  voltage  Ei. 

For  preliminary  calculations  it  is,  therefore,  usually 
permissible  to  substitute  the  terminal  voltage  for  the 
induced  voltage,  and  write  for  formula  (i) 

Ep=       ■'  ^ — -  (approximately).    .     .     (la) 
Similarly, 

^^^4A4f^Ts  (approximately),    .     .     (ib) 

where  Es  and  Ts  stand  respectively  for  the  secondary 
terminal  voltage  and  the  number  of  turns  in  secondary. 
It  follows  that, 

Es     Ts'  ^^ 

which  is  approximately  true  in  all  well-designed  static 
transformers  when  no  current,  or  only  a  very  small 
current,  is  taken  from  the  secondary. 

3.  Effect  of  Closing  the  Secondary  Circuit.  When 
considering  the  action  of  a  transformer  with  loaded 
secondary,  that  is  to  say,  with  current  taken  from  the 
secondary  terminals,  it  is  necessary  to  bear  in  mind  that 
— except  for  the  small  voltage  drop  due  to  ohmic  resist- 
ance of  the  primary  winding — the  counter  e.m.f.  induced 


^LEMFNTARY  THEORY— TYPES— CONSTRUCTION       7 

by  the  alternating  magnetic  flux  in  the  core  must  still 
be  such  as  to  balance  the  e.m.f.  impressed  at  primary 
terminals.  It  follows  that,  with  constant  line  voltage, 
the  flux  $  has  very  nearly  the  same  value  at  full  load  as 
at  no  load.  The  m.m.f.  due  to  the  current  in  the  sec- 
ondary windings  would  entirely  alter  the  magnetization 
of  the  core  if  it  were  not  immediately  counteracted  by 
a  current  component  in  the  primary  windings  of  exactly 
the  same  magnetizing  effect,  but  tending  at  every  instant 
to  set  up  flux  in  the  opposite  direction.  Thus,  in  order 
to  maintain  the  flux  necessary  to  produce  the  required 
counter  e.m.f.  in  the  primary,  any  tendency  on  the  part 
of  the  secondary  current  to  alter  this  flux  is  met  by  a 
flow  of  current  in  the  primary  circuit;  and  since,  in 
well-designed  transformers,  the  magnetizing  current  is 
always  a  small  percentage  of  the  full-load  current,  it 
follows  that  the  relation 

hT,=LTs, (3) 

is  approximately  correct. 

Thus,  ^-f^^, 

where  Ip  and  L  stand  respectively  for  the  total  primary 
and  secondary  current. 

The  open-circuit  conditions  are  represented  in  Fig.  3 
where  Ep  is  the  curve  of  primary  impressed  e.m.f.  and 
le  is  the  magnetizing  current,  distorted  by  the  hysteresis 
of  the  iron  core,  as  will  be  explained  later.     Es  is  the 


8  PRINCIPLES  OF  TRANSFORMER  DESIGN 

curve  of  secondary  e.m.f.  which  coincides  in  phase  with 
the  primary  induced  e.m.f.  and  is  therefore— if  we 
neglect  the  small  voltage  drop  due  to  ohmic  resistance 
of  the  primary— exactly  in  opposition  to  the  impressed 
e.m.f.    The  curve  of  magnetization  (not  shown)  would 


Fig.  3. — Voltage  and  Current  Cun'es  of  Transformer  with  Open  Second- 
ary Circuit. 


be  exactly  a  quarter  period  in  advance  of  the  induced, 
or  secondary,  e.m.f. 

In  Fig.  4,  the  secondary  circuit  is  supposed  to  be  closed 
on  a  non-inductive  load,  and  the  secondary  current, 
/,  will,  therefore,  be  in  phase  with  the  secondary 
e.m.f. 


ELEMENTARY  THEORY— TYPES— CONSTRUCTION       9 

The  tendency  of  the  secondary  current  being  to  pro- 
duce a  change  in  the  magnetization  of  the  core,  the  cur- 
rent in  the  primary  will  immediately  adjust  itself  so  as 
to  maintain  the  same  (or  nearly  the  same)  cycle  of  mag- 
netization as  pn  open  circuit;    that  is  to  say,  the  flux 


Fig.  4. — Voltage  and  Current  Curves  of  Transformer  on  Non-inductive 
Load. 


will  continue  to  be  such  as  will  produce  an  e.m.f.  in  the 
primary  windings  equal,  but  opposite,  to  the  primary 
impressed  potential  difference.  The  new  curve  of  pri- 
mary current,  Ij,  (Fig.  4) ,  is  therefore  obtained  by  adding 
the  ordinates  of  the  current  curve  of  Fig.  3  to  those  of 
another  curve  exactly  opposite  in  phase  to  the  secondary 


10 


PRINCIPLES  OF  TRANSFORMER  DESIGN 


current,  and  of  such  a  value  as  to  produce  an  equal  mag- 
netizing effect. 

4.  Vector  Diagrams  of  Loaded  Transformer  Without 
Leakage.  The  diagram  of  a  transformer  with  secondary 
closed  on  a  non-inductive  load  is  shown  in  Fig.  5.  In 
order  to  have  a  diagram  of  the  simplest  kind,  not  only 
the  leakage  flux,  but  also  the  resistance  of  the  windings 


e;        I:  ,  I„  0  I.        E, 

Fig.  5. — Vector  Diagram  of  Transformer  on  Non-inductive  Load. 


will  be  considered  negUgible.     The  vectois  then  have  the 
following  meaning: 

05  =  Phase  of  flux  $  linked  with  both  primary  and 

secondary  windings; 
7e  =  Exciting  current  necessary  to  produce  flux  <I>; 
£2  =  Secondary  e.m.f.   produced  by  alternations  of 

the  flux  $; 
£'1  =  Primary    e.m.f.    equal,    but    opposite,    to    the 

e.m.f.  produced  by  alternations  of  the  flux  * 

(In  this  case  it  is  equal  to  the  applied  e.m.f., 

since  the  IR  drop  is  negligible) ; 


ELEMENTARY  THEORY— TYPES— CONSTRUCTION     11 

Is  =  Current  drawn  from  secondary;  in  phase  with  £2 ; 

/i=  Balancing  component  of  primary  current,  drawn 

Ts 
exactly  opposite  to  L  and  of  value  /,X;~; 

/,  =  Total  primary  current,  obtained  by  combining 
/i  with  le. 

In  Fig.  6  the  vectors  have  the  same  meaning  as  above, 
but  the  load  is  supposed  to  be  partly  inductive,  which 
accounts  for  the  lag  of  /«  behind  £2. 


e; 


E. 


Fig.  6. — Vector  Diagram  of  Transformer  on  Inductive  Load. 

It  is  convenient  in  vector  diagrams  representing  both 
primary  and  secondary  quantities  to  assume  a  i  :  i 
ratio  in  order  that  balancing  vectors  may  be  drawn 
of  equal  length.  The  voltage  vectors  may,  if  preferred, 
be  considered  as  volts  per  turn,  while  the  secondary 
current  vector  can  be  expressed  in  terms  of  the  pri- 
mary current  by  multiplying  the  quantity  representing 

Ts 
the  actual  secondary  current  by  the  ratio  7^. 


12  PRINXIPLES  OF  TRANSFORMER  DESIGN 

5.  Polyphase  Transformers.  Although  we  have  con- 
sidered only  the  smgle-phase  transformer,  all  that  has 
been  said  applies  also  to  the  polyphase  transformer 
because  each  limb  can  be  considered  separately  and 
treated  as  if  it  were  an  independent  single-phase  trans- 
former. 

In  practice  it  is  not  unusual  to  use  single-phase  trans- 
formers on  polyphase  systems,  especially  when  the  units 
are  of  very  large  size.  Thus,  in  the  case  of  a  three-phase 
transmission,  suppose  it  is  desired  to  step  up  from  6600 
volts  to  100,000  volts,  three  separate  single-phase  trans- 
formers can  be  used,  with  windings  grouped  either  Y 
or  A,  and  the  grouping  on  the  secondary  side  need  not 
necessarily  be  the  same  as  on  the  primary  side.  A 
saving  in  weight  and  first  cost  may  be  effected  by  com- 
bining the  magnetic  circuits  of  the  three  transformers 
into  one.  There  would  then  be  three  laminated  cores 
each  wound  with  primary  and  secondary  coils  and  joined 
together  magnetically  by  suitable  laminated  yokes; 
but  since  each  core  can  act  as  a  return  circuit  for  the 
flux  in  the  other  two  cores,  a  saving  in  the  total  weight 
of  iron  can  be  effected.  Except  for  the  material  in  the 
yokes,  this  saving  is  similar  to  the  saving  of  copper  in 
a  three-phase  transmission  line  using  three  conductors 
only  (as  usual)  instead  of  six,  as  would  be  necessary 
if  the  three  single-phase  circuits  were  kept  separate. 
In  the  case  of  a  two-phase  transformer,  the  windings 
would  be  on  two  limbs,  and  the  common  hmb  for  the 
return  flux  need  only  be  of  sufficient  section  to  carry 
V2  times  the  flux  in  any  one  of  the  wound  limbs. 

It  is  not  always  desirable  to  effect  a  saving  in  first 


ELEMENTARY  THEORY— TYPES— CONSTRUCTION     13 

cost  by  installing  polyphase  tiansformers  in  place  of 
single-phase  units,  especially  in  the  large  sizes,  because, 
apart  from  the  increased  weight  and  difficulty  in  hand- 
ling the  polyphase  transformer,  the  use  of  single-phase 
units  sometimes  leads  to  a  saving  in  the  cost  of  spares  to 
be  carried  in  connection  with  an  important  power  devel- 
opment. It  is  unusual  for  all  the  circuits  of  a  polyphase 
system  to  break  down  simultaneously,  and  one  spare 
single-phase  transformer  might  be  sufficient  to  prevent 
a  serious  stoppage,  while  the  repair  of  a  large  polyphase 
transformer  is  necessarily  a  big  undertaking. 

6.  Problems  in  Design.  The  volt-ampere  input  of  a 
single-phase  transformer  is  Epip,  and  if  we  substitute 
for  Ep  the  value  given  by  formula  (la),  we  have 

Volt-amperes  =       g  X  $  X  Tplp. 

Thus,  for  a  given  flux  $,  which  will  determine  the  cross- 
section  of  the  iron  core,  there  is  a  definite  number  of 
ampere  turns  which  will  determine  the  cross-section  of 
the  winding  space.  There  is  no  limit  to  the  number  of 
designs  which  will  satisfy  the  requirements  apart  from 
questions  of  heating  and  efficiency;  but  there  is  obvi- 
ously a  relation  between  the  weight  of  iron  and  weight 
of  copper  which  will  produce  the  most  economical 
design,  and  this  point  will  be  taken  up  when  discussing 
procedure  in  design.  It  will,  however,  be  necessary  to 
consider,  in  the  first  place,  a  few  practical  points  in 
connection  with  the  construction  of  transformers,  and 
also  the  effect  of  insulation  on  the  space  available  for  the 
copper.    The  predetermination  of  the  losses  in  both  iron 


14  PRINCIPLES  OF  TRANSFORMER  DESIGN 

and  copper  must  then  be  studied  with  a  view  to  calcu- 
lating the  temperature  rise  and  efficiency.  Finally,  the 
flux  leakage  must  be  determined  with  a  reasonable 
degree  of  accuracy  because  this,  together  with  the  ohmic 
resistance  of  the  windings,  will  influence  the  voltage 
regulation,  which  must  usually  be  kept  within  specified 
limits. 

7.  Classification  of  Alternating-current  Transformers. 
Since  we  are  mainly  concerned  with  so-called  constant- 
potential  transformers  as  used  on  power  and  lighting 
circuits,  we  shall  not  at  present  consider  constant-current 
transformers  as  used  on  series  lighting  systems  and  in 
connection  with  current-measuring  instruments;  neither 
shall  we  discuss  in  this  place  the  various  modifications 
of  the  normal  t^pe  of  transformer  which  render  it  avail- 
able for  many  special  purposes. 

Transformers  might  be  classified  according  to  the 
method  of  cooling,  or  according  to  the  voltage  at  the 
terminals,  or,  again,  according  to  the  number  of  phases 
of  the  system  on  which  they  will  have  to  operate. 

Methods  of  cooling  will  be  referred  to  again  later  when 
treating  of  losses  and  temperature  rise;  but,  briefly 
stated,  they  include: 

(i)  Natural  cooling  by  air. 

(2)  Self-cooling  by  oil;  whereby  the  natural  circula- 
tion of  the  oil  in  which  the  transformer  is  immersed  car- 
ries the  heat  to  the  sides  of  the  containing  tank. 

(3)  Cooling  by  water  circulation :  a  method  generally 
similar  to  (2)  except  that  coils  of  pipe  carrying  running 
water  are  placed  near  the  top  of  the  tank  below  the 
surface  of  the  oil. 


ELEMENTARY  THEORY— TYPES— CONSTRUCTION     15 

(4)  Cooling  with  forced  circulation  of  oil:  a  method 
used  sometimes  when  coohng  water  is  not  available.  It 
permits  of  the  oil  being  passed  through  external  pipe 
coils  having  a  considerable  heat-radiating  surface. 

(5)  Cooling  by  air  blast;  whereby  a  continuous  stream 
of  cold  air  is  passed  over  the  heated  surfaces,  exactly 
as  in  the  case  of  large  turbo-generators. 

In  regard  to  difference  of  voltage,  this  is  mainly  a 
matter  of  insulation,  which  will  be  taken  up  in  Chap. 
II.  The  essential  features  of  a  potential  transformer 
are  the  same  whether  the  potential  difference  at  ter- 
minals is  large  or  small,  but  the  high-pressure  trans- 
former will  necessarily  occupy  considerably  more  space 
than  a  low-pressure  transformer  of  the  same  k.v.a. 
output.  The  difficulties  of  avoiding  excessive  flux  leak- 
age and  consequent  bad  voltage  regulation  are  increased 
with  the  higher  voltages. 

Low-voltage  transformers  are  used  for  welding  metals 
and  for  any  purpose  where  very  large  currents  are  nec- 
essary, as  for  instance,  in  thawing  out  frozen  water 
pipes,  while  transformers  for  the  highest  pressures  are 
used  for  testing  insulation.  Testing-transformers  to  give 
up  tp  500,000  volts  at  secondary  terminals  are  not 
uncommon,  while  one  transformer  (at  the  Panama- 
Pacific  Exposition  of  191 5)  was  designed  for  an  output 
of  1000  k.v.a.  at  1,000,000  volts.  This  transformer 
weighed  32,000  lb.,  and  225  bbl.  of  oil  were  required  to 
fill  the  tank  in  which  it  was  immersed. 

A  classification  of  transformers  by  the  number  of 
phases  would  practically  resolve  itself — so  far  as  present- 
day  tendencies  are  concerned — into  a  division  between 


16  PRINCIPLES  OF  TRANSFORMER  DESIGN 

single-phase  and  three-phase  transformers.  From  the 
point  of  view  of  the  designer,  it  will  be  better  to  consider 
the  use  to  which  the  transformer— whether  single-phase 
or  polyphase — will  be  put.  This  leads  to  the  two 
classes: 

(i)  Power  transformers. 

(2)  Distributing  transformers. 

Power  Transjormers.  This  term  is  here  used  to 
include  all  transformers  of  large  size  as  used  in  central 
generating  stations  and  sub-stations  for  transforming 
the  voltage  at  each  end  of  a  power  transmission  line. 
They  may  be  designed  for  maximum  efficiency  at  full 
load,  because  they  are  usually  arranged  in  banks,  and 
can  be  thrown  in  parallel  with  other  imits  or  discon- 
nected at  will.  Artificially  cooled  transformers  of  the 
air-blast  type  are  easily  built  in  single  units  for  outputs 
of  3000  k.v.a.  single-phase  and  6000  k.v.a.  three-phase; 
but  the  terminal  pressure  of  these  transformers  rarely 
exceeds  33,000  volts.  A  three-phase  unit  of  the  air- 
blast  type  with  14,000  volts  on  the  high-tension  wind- 
ings has  actually  been  built  for  an  output  of  20,000 
k.v.a.  For  higher  voltages  the  oil  insulation  is  used, 
generally  with  water  cooling-pipes.  These  transformers 
have  been  built  three-phase  up  to  10,000  k.v.a.  output 
from  a  single  unit,  for  use  on  transmission  systems  up 
to  150,000  volts.*     With  the  modern  demand  for  larger 

*The  10,000  k.v.a.  three-phase,  6600  to  110,000-volt  units  in  the 
power  houses  of  the  Tennessee  Power  Company  on  the  Ocoee  River  weigh 
about  200,000  lb.;  they  are  19  ft.  high,  and  occupy  a  floor  space  20  ft. 
by  8  ft. 

Single-phase,  oil-insulated,  water-cooled  transformers  for  a  frequency 
of  60  cycles  and  a  ratio  of  13,200  to  150,000  volts  have  been  built  for  an 
output  of  14,000  k.v.a.  from  a  single  unit. 


ELEMENTARY  THEORY— TYPES— CONSTRUCTION     17 

transformers  to  operate  out  of  doors,  power  transformers 
of  the  oil-immersed  self-cooling  type  (without  water 
coils)  are  now  being  constructed  in  increasing  number. 
A  self-cooling  25-cycle  transformer  for  8000  k.v.a.  out- 
put has  actually  been  built:  a  number  of  special  tube- 
type  radiators  connected  by  pipes  to  the  main  oil  tank 
are  provided;  the  total  cooling  surface  in  contact  with 
the  air  being  about  7000  sq.  ft. 

Distributing  Transformers.  These  are  always  of  the 
self-cooling  type,  and  almost  invariably  oil-immersed. 
They  include  the  smaller  sizes  for  outputs  of  i  to  3  k.w. 
such  as  are  commonly  mounted  on  pole  tops.  These 
transformers  are  rarely  wound  for  pressures  exceeding 
13,000  volts,  the  most  common  primary  voltage  being 
2200. 

In  the  design  of  distributing  transformers,  it  is  neces- 
sary to  bear  in  mind  that  since  they  are  continuously 
on  the  circuit,  the  "  all-day  ''*  losses — which  consist 
largely  of  hysteresis  and  eddy-current  losses  in  the  iron — 
must  be  kept  as  small  as  possible.  In  other  words,  it  is 
not  always  desirable  to  have  the  highest  efficiency  at 
full  load. 

8.  Types  of  Transformers.  Construction.  All  trans- 
formers consist  of  a  magnetic  circuit  of  laminated  iron 
with  which  the  electric  circuits  (primary  and  secondary) 
are  linked.  A  distinction  is  usually  made  between  core- 
type  and  shell-type  transformers.  Single-phase  trans- 
formers of  the  core-  and  shell-t>'pes  are  illustrated  by 
Figs.  7  and  8,  respectively.  The  former  shows  a  closed 
laminated  iron  circuit  two  hmbs  of  which  carry  the  wind- 
ings.    Each   limb   is   wound   with   both   primary   and 


18 


PRINCIPLES  OF  TRANSFORMER  DESIGN 


secondary  circuits  in  order  to  reduce  the  magnetic  leak- 
age which  would  otherwise  be  excessive.  The  coils  may 
be  cylindrical  in  form  and  placed  one  inside  the  other 
with  the  necessary  insulation  between  them,  or  the  wind- 
ings may  be  "  sandwiched,"  in  which  case  flat  rect- 
angular or  circular  coils,  alternately  primary  and  sec- 


FiG.  7. — Core-type  Transformer.        Fig.  8. — Shell-t)T5e  Transformer, 


ondary,  are  stacked  one  above  the  other  with  the  requi- 
site insulation  between. 

Fig.  8  shows  a  single  set  of  windings  on  a  central 
laminated  core  which  divides  after  passing  through  the 
coils  and  forms  what  may  be  thought  of  as  a  shell  of 
iron  around  the  copper.  The  manner  in  which  the  core 
is  usually  built  up  in  a  large  shell-type  transformer  is 
shown  in  Fig.   9.     The   thickness  of    the  laminations 


ELEMENTARY  THEORY— TYPES— CONSTRUCTION     19 

varies  between  0.012  and  0.018  in.,  the  thicker  plates 
being  permissible  when  the  frequency  is  low.  A  very 
usual  thickness  for  transformers  working  on  25-  and  60- 
cycle  circuits  is  0.014  in.  The  arrangement  of  the 
stampings  is  reversed  in  every  layer  in  order  to  cover 
the  joints  and  so  reduce  the  magnetizing  component 
of  the  primary  current.     A  very  thin  coating  of  varnish 


Z' 

N 

\ 

/ 

y 

\ 

\ 

V 


Fig.  9. — Method  of  Assembling  Stampings  in  Shell-type  Transformer. 


or  paper  is  sufficient  to  afford  adequate  insulation  be- 
tween stampings.  Ordinary  iron  of  good  magnetic 
quality  may  be  used  for  transformers  on  the  lower  fre- 
quencies, but  it  is  customary  to  use  special  alloyed 
iron  for  60-cycle  transformers.  This  material  has  a 
high  electrical  resistance  and,  therefore,  a  small  eddy- 
current  loss.  The  loss  through  hysteresis  is  also  small, 
but  the  permeability  of  alloyed  iron  is  lower  than  that 
of  ordinary  iron  and  this  tends  to  increase  the  magnetiz- 


20  PRINCIPLES  OF  TRANSFORMER  DESIGN 

ing  current.  The  cost  of  alloyed  iron  is  appreciably 
higher  than  that  of  ordinary  transformer  iron. 

The  choice  of  type — whether  "  core  "  or  "  shell  " — 
will  not  greatly  affect  the  efficiency  or  cost  of  the  trans- 
former. As  a  general  rule,  the  core  type  of  construc- 
tion has  advantages  in  the  case  of  high-voltage  trans- 
formers of  small  output,  while  the  shell  type  is  best 
adapted  for  low- voltage  transformers  of  large  output. 

Fig.  ID  illustrates  a  good  practical  design  of  shell-type 
transformer  in  which  a  saving  of  material  is  effected 
by  arranging  the  magnetic  circuit  to  surround  all  four 
sides  of  a  square  coil.  The  dimensions  of  the  iron  cir- 
cuit, as  indicated  on  the  sketch,  show  a  cross-section 
of  the  magnetic  circuit  outside  the  coils  exactly  double 
the  cross-section  inside  the  coils.  This  will  be  found  to 
lead  to  slightly  higher  efficiency,  for  the  same  cost  of 
material,  than  if  the  section  were  the  same  inside  and 
outside  the  coil.  It  is  generally  advantageous  to  use 
higher  flux  densities  in  the  iron  upon  which  the  coils 
are  wound  than  in  the  remainder  of  the  magnetic  cir- 
cuit, because  the  increased  iron  loss  is  compensated  for 
by  the  reduced  copper  loss  due  to  the  shorter  average 
length  per  turn  of  the  windings. 

Fig.  II  illustrates  a  similar  design  of  shell-type  trans- 
former in  which  the  magnetic  circuit  is  still  further 
divided,  and  the  windings  are  in  the  form  of  cylindrical 
coils.  The  relative  positions  of  primary  and  secondary 
coils  need  not  be  as  shown  in  Figs.  lo  and  ii,  as  they 
can  be  of  the  "  pancake  "  shape  of  no  great  thickness, 
with  primary  and  secondary  coils  alternating.  A  proper 
arrangement  of  the  coils  is  a  matter  of  great  importance 


ELEMENTARY  THEORY— TYPES— CONSTRUCTION     21 

when  it  is  desired  to  have  as  small  a  voltage  drop  as 
possible  under  load;    but  this  point  will  be  taken  up 


Fig.  io.— Shell-type  Transformer  with  Distributed  Magnetic  Circuit. 
(Square  core  and  coil.) 


again  when  dealing  with  magnetic  leakage  and  regula- 
tion. 
Fig.   12   illustrates  a  common  arrangement  of  the 


22 


PRINCIPLES  OF  TRANSFORMER  DESIGN 


stampings  and  windings  in  a  three-phase  core-type 
transformer.  Each  of  the  three  cores  carries  both  pri- 
mary and  secondary  coils  of  one  phase.    The  portions 


Fig.  II. — Shell-type  Transformer  with  Distributed  Magnetic  Circuit. 
(Berry  transformer  with  circular  coil.) 


of  the  magnetic  circuit  outside  the  coils  must  be  of 
sufficient  section  to  carry  the  same  amount  of  flux  as 
the  wound  cores.  This  will  be  understood  if  a  vector 
diagram  is  drawn  showing  the  flux  relations  in  the 


ELEMENTARY  THEORY— TYPES— CONSTRUCTION     23 

various  parts  of  the  rnagnetic  cir(!uit.  This  use  of  cer- 
tain parts  of  the  magnetic  circuit  to  carry  the  flux  com- 
mon to  all  the  cores  leads  to  a  saving  in  material  on 
what  would  be  necessary  for  three  single-phase  trans- 
formers of  the  same  total  k.v.a.  output;   but,  as  men- 


FiG.  12. — Three-phase  Core-type  Transformer. 


tioned  in  Article  5,  it  does  not  follow  that  a  three-phase 
transformer  is  always  to  be  preferred  to  three  separate 
single-phase  transformers. 

Figs.  13  and  14  show  sections  through  three-phase 
transformers  of  the  shell  type.    The  former  is  the  more 


24 


PRINCIPLES  OF  TRANSFORMER  DESIGN 


common  design,  and  it  has  the  advantage  that  rect- 
angular shaped  stampings  can  be  used  throughout. 
The  vector  diagram  in  Fig,  13  shows  how  the  flux  $« 
in  the  portion  of  the  magnetic  circuit  between  two  sets 
of  coils  has  just  half  the  value  of  the  flux  $  in  the  cen- 
tral core. 


Fig.    13. — Section    through    Three-phase    Shell    Transformer.     (Each 
phase  consists  of  one  H.T.  and  two  L.T.  coils.) 

9.  Mechanical  Stresses  in  Transformers.  The 
mechanical  features  of  transformer  design  are  not  of 
sufficient  importance  to  warrant  more  than  a  brief 
discussion.  In  the  smaller  transformers  it  is  merely 
necessary  to  see  that  the  clamps  or  frames  securing 
the  stampings  and  coils  in  position  are  sufficiently  sep- 
arated from  the  H.T.  windings,  and  that  bolts  in  which 


H,  C.  Stot«  Collttt 


ELEMENTARY  THEORY— TYPES— CONSTRUCTION     25 

e.m.f.'s  are  likely  to  be  generated  by  the  main  or  stray 
magnetic  fluxes  are  suitably  insulated  to  prevent  the 
establishment  of  electric  currents  with  consequent  PR 
losses.  The  tendency  in  all  modern  designs  is  to  avoid 
cast  iron,  and  use  standard  sections  of  structural  steel 
in  the  assembly  of  the  complete  transformer.     In  this 


Fig.  14. — Special  Design  of  Three-phase  Shell-type  Transformer. 


manner  the  cost  of  special  patterns  is  avoided  and  a 
saving  in  weight  is  usually  effected.  The  use  of  stand- 
ard steel  sections  also  gives  more  flexibiUty  in  design, 
as  sKght  modifications  can  be  made  in  dimensions  with 
very  Uttle  extra  cost. 

\  In  large  transformers,  the  magnetic  forces  exerted 
under  conditions  of  heavy  overloads  or  short-circuits 


26  PRINCIPLES  OF  TRANSFORMER  DESIGN 

may  be  sufficient  to  displace  or  bend  the  coils  unless 
these  are  suitably  braced  and  secured  in  position;  and 
since  the  calculation  of  the  stresses  that  have  to  be 
resisted  belong  properly  to  the  subject  of  electrical 
design,  it  will  be  necessary  to  determine  how  these 
stresses  can  be  approximately  predetermined. 

The  absolute  unit  of  current  may  be  defined  as  the 
current  in  a  wire  which  causes  one  centimeter  length 
of  the  wire,  placed  at  right  angles  to  a  magnetic  field, 
to  be  pushed  sidewise  with  a  force  of  one  dyne  when 
the  density  of  the  magnetic  field  is  one  gauss. 

Since  the  ampere  is  one-tenth  of  the  absolute  unit  of 
current,  we  may  write, 

where  F  =  Force  in  dynes ; 

5  =  Density  of  the  magnetic  field  in  gausses; 
/  =  Current  in  the  wire  (amperes); 
/  =  Length  of  the  wire  (centimeters)  in  a  direction 
perpendicular  to  the  magnetic  field. 

It  follows  that  the  force  tending  to  push  a  coil  of  wire 
of  T  turns  bodily  in  a  direction  at  right  angles  to  a 
uniform  magnetic  field  of  B  gausses  (see  Fig.  1 5)  is 

F  = dynes. 

10 

If  both  current  and  magnetic  field  are  assumed  to 
vary  periodically  according  to  the  sine  law,  passing 
through   corresponding   stages   of   their   cycles  at   the 


ELEMENTARY  THEORY— TYPES— CONSTRUCTION     27 


same  instant  of  time,  we  have  the  condition  which  is 
approximately  reproduced  in  the  practical  transformer 
where  the  leakage  flux  passing  through  the  windings  is 
due  to  the  currents  in  these  windings. 


Uniform  Field 
of  B  gausses 


Coil  of  T  wires,  eaoh 
carrying  I   amperes 


Fig.  15. — Force  Acting  on  Coil-side  in  Uniform  Magnetic  Field. 

Since  the  instantaneous  values  of  the  current  and 
flux  density  will  be  I^^^  sin  6,  and  B^^^^  sin  6,  respectively, 
the  average  mechanical  force  acting  upon  the  coil  may 
be  written, 

'pjT  n 

Favera<.e  =  — /max-Bmax-  I      SIR^  Odd  = 


if 


10X2 


dynes. 


28  PRINCIPLES  OF  TRANSFORMER  DESIGN 

If  the  flux  density  is  not  uniform  throughout  the  sec- 
tion of  coil  considered,  the  average  value  of  ^niax  should 
be  taken.  Let  this  average  value  of  the  maximum  den- 
sity be  denoted  by  the  symbol  B^rn-  Then,  since  i  lb.= 
444,800  dynes,  the  final  expression  for  the  average 
force  tending  to  displace  the  coil  is, 

Average  force  =  ^^^7^-"  lb.  .     .     .     (4) 
8,896,000 

In  large  transformers  the  amount  of  leakage  flux 
passing  through  the  coils  may  be  considerable.  It  will 
be  very  nearly  directly  proportional  to  1^^^,  and  the 
mechanical  forces  on  transformer  coils  are  therefore 
approximately  proportional  to  the  square  of  the  current. 
As  the  short-circuit  current  in  a  transformer  which  is 
not  specially  designed  with  high  reactance  might  be 
thirty  times  the  normal  full-load  current,  the  mechan- 
ical forces  due  to  a  short-circuit  may  be  about  1000 
times  as  great  as  the  forces  existing  under  normal  work- 
ing conditions. 

Except  in  a  few  special  cases,  the  calculation  of  the 
leakage  flux  is  not  an  easy  matter,  and  the  value  of  Bf^ja 
in  Eq.  (4)  cannot  usually  be  predetermined  exactly; 
but  it  can  be  estimated  with  sufficient  accuracy  for  the 
purpose  of  the  designer,  who  requires  merely  to  know 
approximately  the  magnitude  of  the  mechanical  forces 
which  have  to  be  resisted  by  proper  bracing  of  the  coils. 

The  calculation  of  leakage  flux  will  be  considered 
when  discussing  voltage  regulation;  but  in  the  case  of 
"sandwiched  "  coils  as,  for  instance,  in  the  shell  type  of 


ELEMENTARY  THEORY— TYPES— CONSTRUCTION     29 

transformer  shown  in  Fig.  i6,  the  distribution  of  the 
leakage  flux  will  be  generally  as  indicated  by  the  dia- 
gram plotted  over  the  coils  at  the  bottom  of  the  sketch. 


Fig.  1 6.— Forces  in  Transformer  Coils  Due  to  Leakage  Flux. 


When  the  relative  directions  of  the  currents  in  the 
primary  and  secondary  coils  are  taken  into  account,  it 


30 


PRINCIPLES  OF  TRANSFORMER  DESIGN 


will  be  seen  that  all  the  forces  tending  to  push  the  coils 
sidewise  are  balanced,  except  in  the  case  of  the  two 
outside  coils.  In  each  individual  coil  the  effect  of  the 
leakage  flux  is  to  crush  the  wires  together;  but  the  end 


Fig.  17. — Core-type  Transformer  with  "Sandwiched"  Coils. 


coils  will  be  pushed  outward  unless  properly  secured  in 
position. 

Since  there  is  no  resultant  force  tending  to  move  the 
windings  bodily  relatively  to  the  iron  stampings,  a 
simple  form  of  bracing  consisting  of  insulated  bars  and 


ELEMENTARY  THEORY— TYPES— CONSTRUCTION     31 

tie  rods,  as  shown  in  Fig.  i6  will  satisfy  all  requirements, 
and  this  bracing  can  be  quite  independent  of  the  frame- 
work or  clamps  supporting  the  transformer  as  a  whole. 
In  the  case  of  core-type  transformers,  with  rect- 
angular coils  arranged  axially  one  within  the  other,  the 
mechanical  forces  will  tend  to  force  the  coils  into  a  cir- 
cular shape.  With  cylindrical  concentric  coils,  no  spe- 
cial bracing  is  necessary  provided  the  coils  are  symmet- 
rically placed  axially;  but  if  the  projection  of  one  coil 
beyond  the  other  is  not  the  same  at  both  ends,  there 
will  be  an  unbalanced  force  tending  to  move  one  coil 
axially  relatively  to  the  other.  If  the  core  type  of 
transformer  is  built  up  with  flat  strip  "  sandwiched  " 
coils,  the  problem  is  generally  similar  to  that  of  the  shell 
type  of  construction.  A  method  of  securing  the  end  coils 
in  position  with  this  arrangement  of  windings  is  illus- 
trated by  Fig.  17. 


CHAPTER  II 

INSULATION  OF  HIGH-PRESSURE  TRANSFORMERS 

10.  The  Dielectric  Circuit.  Serious  difficulties  are  not 
encountered  in  insulating  machinery  and  apparatus 
for  working  pressures  up  to  10,000  or  12,000  volts,  but 
for  higher  pressures  (as  in  150,000- volt  transformers) 
designers  must  have  a  thorough  understanding  of  the 
dielectric  circuit,*  if  the  insulation  is  to  be  correctly 
and  economically  proportioned.  The  information  here 
assembled  should  make  the  fundamental  principles  of 
insulation  readily  understood  and  should  enable  an 
engineer  to  determine  in  any  specific  design  of  trans- 
former the  thicknesses  of  insulation  required  in  any 
particular  position,  as  between  layers  of  windings, 
between  high-tension  and  low- tension  coils,  and  be- 
tween high-tension  coils  and  grounded  metal.  The  data 
and  principles  outKned  should  also  faciHtate  the  deter- 
mination of  dimensions  and  spacings  of  high-tension 
terminals  and  bushings  of  which  the  detailed  design  is 
usually  left  to  speciaKsts  in  the  manufacture  of  high- 
tension  insulators.  In  presenting  this  information  two 
questions  are  considered:     (i)   What  is  the  dielectric 

* "  Insulation  and  Design  of  Electrical  Windings,"  by  A.  P.  M 
Fleming  and  R.  Johnson — Longmans,  Green  &  Co. 

"  Dielectric  Phenomena  in  High-voltage   Engineering,"  by  F.  W. 
Peek,  Jr. — McGraw-Hill  Book  Company,  Inc. 
32 


INSULATION  OF  HIGH-PRESSURE  TRANSFORMERS       33 

strength  of  the  insulating  materials  used  in  transformer 
design?  and  (2)  how  can  the  electric  stress  or  voltage 
gradient  be  predetermined  at  all  points  where  it  is  hable 
to  be  excessive? 

Apart  from  a  few  simple  problems  of  insulation 
capable  of  a  mathematical  solution,  the  chief  difficulty 
encountered  in  practice  usually  Hes  in  determining  the 
distribution  of  the  dielectric  flux,  the  concentration  of 
which  at  any  particular  point  may  so  increase  the  flux 
density  and  the  corresponding  electric  stress  that  dis- 
ruption of  the  dielectric  may  occur.  The  conception  of 
Unes  of  dielectric  flux,  and  the  treatment  of  the  dielec- 
tric circuit  in  the  manner  now  famihar  to  all  engineers 
in  connection  with  the  magnetic  circuit  has  made  it  pos- 
sible to  treat  insulation  problems  *  in  a  way  that  is 
equally  simple  and  logical. 

The  analogy  between  the  dielectric  and  magnetic 
circuits  may  be  illustrated  by  Fig.  18,  where  a  metal 
sphere  is  supposed  to  be  placed  some  distance  away  from 
a  flat  metal  plate,  the  intervening  space  being  occupied 
by  air,  oil,  or  any  insulating  substance  of  constant 
specific  capacity.  This  arrangement  constitutes  a  con- 
denser of  which  the  capacity  is  (say)  C  farads.  If  a 
difference  of  potential  of  E  volts  is  estabUshed  between 

*  The  dielectric  circuit  is  well  treated  from  this  point  of  view  in 
the  following  (among  other)  books: 

"  The  Electric  Circuit,"  by  V.  Karapetoff— McGraw-Hill  Book 
Company,  Inc. 

"  Electrical  Engineering,"  by  C.  V.  Christie— McGraw-Hill  Book 
Company,  Inc. 

"  Advanced  Electricity  and  Magnetism,"  by  W.  S.  Franklin  and 
B.    MacNutt — Macmillan  Company. 


34  PRINCIPLES  OF  TRANSFORMER  DESIGN 

the  sphere  and  the  plate,  the  total  dielectric  flux, 
^  will  have  to  satisfy  the  equation 

^  =  £C,        (5) 

where  ^  is  expressed  in  coulombs,  E  in  volts,  and  C  in 
farads. 

The  quantity  ^  coulombs  of  electricity  should  not  be 
considered  as  a  charge  which  has  been  carried  from  the 
sphere  to  the  plate  on  the  surface  of  which  it  remains, 
because  the  whole  of  the  space  occupied  by  the  dielectric 
is  actually  in  a  state  of  strain,  like  a  deflected  spring, 
ready  to  give  back  the  energy  stored  in  it  when  the 
potential  difference  causing  the  deflection  or  displace- 
ment is  removed.  Instead,  the  dielectric  should  be 
considered  as  an  electrically  elastic  material  which  will 
not  break  down  or  be  ruptured  until  the  "  elastic  limit '' 
has  been  reached.  The  quantity  ^,  which  is  called  the 
dielectric  flux,  may  be  thought  of  as  being  made  up  of 
a  definite  number  of  unit  tubes  of  induction,  the  direc- 
tion of  which  in  the  various  portions  of  the  dielectric 
field  is  represented  by  the  full  lines  in  Fig.  18.  The 
name  of  the  unit  tube  of  dielectric  flux  is  the  coulomb. 

If  the  sphere  were  the  north  pole  and  the  plate  the 
south  pole  of  a  magnetic  circuit,  the  distribution  of 
flux  lines  would  be  similar.  The  total  flux  would  then 
be  denoted  by  the  symbol  <l>,  and  the  unit  tube  of  induc- 
tion would  be  called  the  maxwell.  In  place  of  formula 
(5)  the  following  well-known  equation  could  then  be 
written : 

*=ilfw/X  permeance (6) 


INSULATION  OF  HIGH-PRESSURE  TRANSFORMERS        35 

This  expression  is  analogous  to  the  fundamental 
equation  for  a  dielectric  circuit,  the  electrostatic  capacity 
C  being,  in  fact,  a  measure  of  the  permeance  of  the  di- 
electric circuit,  while  — ,  sometimes  called  the  elastance, 

may   be   compared   with   reluctance   in   the   magnetic 
circuit. 

The  dotted  lines  in  Fig.  i8  are  sections  through  equi- 
potential   surfaces.     The  potential   difference  between 


Fig.  i8. — Distribution  of  Dielectric  Flux  between  Sphere  and  Flat  Plate. 


any  two  neighboring  surfaces,  as  drawn,  is  one-quarter 
of  the  total.  At  all  points  the  lines  of  force,  or  unit 
tubes  of  induction,  are  perpendicular  to  the  equipoten- 
tial  surfaces.  Furthermore,  the  flux  density,  or  cou- 
lombs per  square  centimeter,  through  any  small  portion 
A  of  an  equipotential  surface  over  which  the  distribu- 
tion may  be  considered  practically  uniform  is 


D  = 


^ 


(7) 


36  PRINCIPLES  OF  TRANSFORMER  DESIGN 

The  capacity,  or  permittance*  of  a  small  element  of 

the  dielectric  circuit  of  length  /  and  cross-section  A 

A 
is   proportional   to   — ,   or   with   the  proper  constants 

inserted, 

/        lo^        \kA 
Electrostatic  capacity  =  C  =  (  — 7- 57^  l-r-  farads  (8) 


wherein  the  numerical  multiplier  results  from  the  choice 
of  units.  The  factor  k  is  the  specific  inductive  capacity, 
or  dielectric  constant,  of  the  material  (^  =  1  in  air), 
while  the  unit  for  /  and  A  is  the  centimeter.  This  ex- 
pression for  capacity  may  conveniently  be  rewritten  as 

Cm/=-^  —r  microfarads.       ...     (9) 

Values  of  k  are  given  in  the  accompanying  table  to- 
gether with  the  dielectric  strengths  of  the  materials. 
These  figures  are  only  approximate,  those  referring  to 
dielectric  strength  merely  serving  as  a  rough  indication 
of  what  the  material  of  avei age  quality  may  be  expected 
to  withstand.  The  figures  indicate  the  approximate 
virtual  or  r.m.s.  value  of  the  sinusoidal  alternating 
voltage  which,  if  applied  between  two  large  flat  elec- 
trodes, would  lead  to  the  breakdown  of  a  i-cm.  slab 
of  insulating  material  placed  between  the  electrodes. 

What  is  generally  understood  by  the  disruptive  gra- 
dient, or  stress  in  kilovolts  per  centimeter,  would  be 

*  The  reciprocal  of  elastance. 


INSULATION  OF  HIGH-PRESSURE  TRANSFORMERS       37 

about  \/2  times  the  value  given  in  the  last  column  of 
the  table.  Thus,  if  a  battery  or  continuous-current 
generator  were  used  in  the  test,  the  pressure  necessary  to 
break  down  a  0.75-cm.  film  of  air  between  two  large  flat 
parallel  plates  would  be  ioooX\/2X 22X0.75  =  23,400 
volts. 

Dielectric  Constant  and  Dielectric  Strength  of 
Insulators 


Material. 

Dielectric 
Constant, 

Dielectric 

Strength 

Kv.  per  Cm. 

Air 

I 
2.4 

2  to   2.5 

3  to  4 

3 
S 

3  to  6 

45 

5 
5  to  7 
5  to  7 
5  to  10 
Infinity 

22 

130 
80 

Paper  (dry) 

Paper  (oil  impregnated)         .      . 

Pressboard  (dry  or  varnished) 

ISO 

70 

no 

Porcelain. 

Varnished  cambric 

Mica 

600 

300 

90 

Glass    .  .           

Conductors 

Returning  to  Formula  (7) ,  let  electric  flux  or  quantity 
oj  electricity,  ^,  be  expressed  in  terms  of  capacity  and 
e.m.f.,  with  a  view  to  determining  the  relation  between 
flux  density  and  electric  stress.  The  Formula  (8)  may 
be  written 


C  =  Kk—  farads, 
L 


38  PRINCIPLES  OF  TRANSFORMER  DESIGN 

where    A'    stands    for    the    numerical    constant.     Sub- 
stituting in  Formula  (5), 


whence 


D  =  KkX 


(?)■ 


Since  y  is  the  potential  gradient,  or  voltage  drop 

per  centimeter,  which  is  sometimes  referred  to  as  the 
electrostatic  force  or  electrifying  force,  and  denoted  by 
the  symbol  G,  we  may  write, 

D  =  KkxG (10) 

The  analogous  expression  for  the  magnetic  circuit  is, 

In  the  case  of  a  dielectric  circuit,  electric  flux  density 
=  e.m.f.  per  centimeter  X  "  conductivity  "  of  the  material 
to  dielectric  flux,  while  in  the  magnetic  circuit,  magnetic 
flux  density  =  m.mf.  per  centimeter  X  "  conductivity  " 
of  the  material  to  magnetic  flux. 

Since  the  electric  stress  or  voltage  gradient  G  is 
directly  proportional  (in  a  given  material)  to  the  flux 
density  D,  it  follows  that  when  the  concentration  of 
the  flux  tubes  is  such  as  to  produce  a  certain  maximum 
density  at  any  point,  breakdown  of  the  insulation  will 
occur  at  this  point.  Whether  or  not  the  rupture  will 
extend  entirely  through  the  insulation  will  depend  upon 


INSULATION  OF  HIGH-PRESSURE  TRANSFORMERS        39 

the  value  of  the  flux  density  (consequently  the  potential 
gradient)  immediately  beyond  the  limits  of  the  local 
breakdown. 

Given  two  electrical  conductors  of  irregular  shape, 
separated  by  insulating  materials,  the  problem  of  cal- 
culating the  capacity  of  the  condenser  so  formed  is 
very  similar  to  that  of  calculating  the  permeance  of 
the  magnetic  paths  between  two  pieces  of  iron  of  very 
high  permeability  separated  by  materials  of  low  per- 
meabihty.  There  is  no  simple  mathematical  solution 
to  such  a  problem,  and  the  best  that  can  be  done  is  to 
fall  back  on  the  well-estabhshed  law  of  maximum  per- 
meance, or  "  least  resistance."  According  to  this  law 
the  lines  of  force  and  equipotential  surfaces  will  be  so 
shaped  and  distributed  that  the  permittance,  or  capacity, 
of  the  flux  paths  will  be  a  maximum.  With  a  little 
experience,  ample  time,  and  a  great  deal  of  patience, 
the  probable  field  distribution  can  generally  be 
mapped  out,  even  in  the  case  of  irregularly  shaped 
surfaces,  with  sufficient  accuracy  to  emphasize 
the  weak  points  of  the  design  and  to  permit  of 
the  maximum  voltage  gradient  being  approximately 
determined.* 

Before  illustrating  the  appHcation  of  the  above  prin- 
ciples in  the  design  of  transformer  insulation,  it  will 
be  advisable  to  assemble  and  define  the  quantities  which 
are  of  interest  to  the  engineer  in  making  practical 
calculations. 

*  This  method  of  plotting  flux  lines  is  explained,  in  connection  with 
the  magnetic  field,  at  some  length  in  the  writer's  book  "  Elements 
of  Electrical  Design." — McGraw-Hill  Book  Co.,  Inc. 


40  PRINCIPLES  OF  TRANSFORMER  DESIGN 

Symbol: 

E,  e  =  e.m.i.  or  potential  difference  (volts); 

/  =  length,  measured  along  line  of  force  (centimeters) ; 

A  =  Area  of   equipotential   surface   perpendicular   to 

lines  of  force  (square  centimeters) ; 

de 
G  =  -^  =  potential  gradient  (volts  per  centimeter); 

C  =  Capacity  or  permittance  (farads) ; 

..       ,      coulombs  V         f\ 

(farads  = ^^ —  =  ^^^  P^^  ^^^^  e.m.f .) ; 

K  =  constant  =  8.84  X  io~^^  (farads  per  centimeter  cube, 
being  the  specific  capacity  of  air) ; 

Jfe  =  dielectric  constant,  or  relative  specific  capacity, 
or  permittivity  (^  =  i  for  air) ; 

^  =  dielectric  flux,  or  electrostatic  induction  (^  = 
CE  =  AD  coulombs) ; 

D  =  flux  density = -j  =  KkG  (coulombs  per  square  centi- 
meter) . 

11.  Capacity  of  Plate  Condenser.  Imagine  two  par- 
allel metal  plates,  as  in  Fig.  19,  connected  to  the  oppo- 
site terminals  of  a  direct-current  generator  or  battery. 
The  area  of  each  plate  is  A  square  centimeters  and  the 
separation  between  plates  is  /  centimeters,  the  dielectric 
or  material  between  the  two  surfaces  being  air.  The 
edges  of  the  plates  should  be  rounded  off  to  avoid  con- 
centration of  flux  Hnes.  If  the  area  A  is  large  in  com- 
parison with  the  distance  /,  a  uniform  distribution  of  the 
flux  ^  may  be  assumed  in  the  air  gap,  the  density  being 


INSULATION  OF  HIGH-PRESSURE  TRANSFORMERS        41 


By  Formula  (9)  the  capacity  is  Cm/=    '   g     .    micro- 

10  /\l' 

farads,  since  the  specific  capacity  of  air  (k)  is  i.     As- 
suming  numerical   values,    let   yl  =  iooo   sq.    cm.,    and 

8.84X1000 

/  =  o.5  cm.     Then,  C  =  — r^— =  1.77X10-1°  farads. 

•^  ioi-*Xo.5  " 

If  £=10,000  volts,    the  potential  gradient  will  be 

10,000 

G  = =  20,000  volts  per  centimeter.     There  will  be 

0.5 


Flat  Electrodes  Separated  by  Air. 


no  disruptive  discharge,  however,  because  a  gradient 
of  31,000  volts  per  centimeter  is  necessary  to  cause 
break-down  in  air. 

By  Formula  (5)  the  total  dielectric  flux  is  '^  =  10,000 
X  1.77X10-1°  =  1,77X10-6  coulombs. 

Charging  Current  with  Alternating  Voltage.  The  effect 
of  an  alternating  e.m.f.,  the  crest  value  of  which  is  10,000 
volts,  would  be  to  displace  the  above  quantity  of  elec- 
tricity 4/  times  per   second,  /  being   the   frequency. 


42  PRINCIPLES  OF  TRANSFORMER  DESIGN 

The  quantity  of  electricity  can  be  expressed  in  terms 
of  current  and  time,  thus,  quantity  =  current  X  time,  or 
coulombs  =  average  value  of  current  {in  amperes)  during 
quarter  period Xtime  {in  seconds)  of  one  quarter  period. 

(2V'2\  I 
I -7,   where   /    stands    for    the 

virtual  or  r.m.s.  value  of  the  charging  current  on  the 

sme  wave   assumption.     Transposmg   terms,   /  =  -—=. 

2V2 

If  E  is  now  understood  to  stand  for  the  virtual  value 
of  the  alternating  potential  difference,  ^  =  C£xV2, 
whence  I  =  2irfCE,  which  is  the  well-known  formula 
for  calculating  capacity  current  on  the  assumption  of 
sinusoidal  wave  shapes. 

12.  Capacities  in  Series.  When  condensers  are  con- 
nected in  parallel  on  the  same  source  of  voltage,  the 
total  dielectric  flux  is  evidently  determined  by  summing 
up  the  fluxes  as  calculated  or  measured  for  the  individual 
condensers.  In  other  words,  the  total  capacity  is  the 
sum  of  the  individual  capacities.  With  condensers  in 
series,  however,  the  total  flux,  or  displacement,  will  be 
the  same  for  all  the  capacities  in  series,  therefore,  the 
calculations  may  be  simplified  just  as  for  electric  or 
magnetic  circuits  by  adding  the  reciprocals  of  the  con- 
ductance or  permeance.  The  conception  of  elastance, 
corresponding  to  resistance  in  the  electric  circuit  and 
reluctance  in  the  magnetic  circuit,  is  thus  seen  to 
have  certain  advantages.     In  the  dielectric  circuit 


Elastance  = r- 7 r-^ = 7;. 

permittance  (or  capacity)     C 


INSULATION  OF  HIGH-PRESSURE  TRANSFORMERS       43 


For  a  concrete  example,  assume  that  a  0.3-cm.  plate 
of  glass  is  inserted  between  the  electrodes  of  the  con- 
denser shown  in  Fig.  19.  The  modified  arrangement  is 
illustrated  by  Fig.  20.  On  first  thought  it  might  appear 
that  this  arrangement  would  improve  the  insulation, 
but  care  must  always  be  taken  when  putting  layers  of 
insulating  materials  of  different  specific  inductive  capac- 
ity in  series,  as  this  example  will  illustrate.  In  addition 
to  the  elastance  of  a  0.3-cm.  layer  of  glass  there  is  the 


1    ^tJUBB 


^Glass.plate  0.3  cm.  thick. 


Fig.  20. — Electrodes  Separated  by  Air  and  Glass. 

elastance  of  two  layers  of  air  of  which  the  total  thickness 
is  0.2  cm.  Assuming  that  the  value  of  the  dielectric 
constant  k  for  the  particular  quahty  of  glass  used  is 
7  and  that  Gg  and  Ga  are  the  potential  gradients  in 
the  glass  and  air  respectively,  then,  by  formula  (lo) 
KG  a  =  tKGo,  whence  Ga  =  ^Gg. 

Taking  the  total  potential  difference  between  elec- 
trodes as  10,000  volts,  the  same  as  used  in  considering 
Fig.  19,  £  =  10,000  =  o.2Ga-fo.3G(„  whence  Gp  =  5880  volts 


44 


PRINCIPLES  OF  TRANSFORMER  DESIGN 


per  centimeter,  and  Ga  =  41,100  volts  per  centimeter. 
Such  a  high  gradient  as  41,100  would  break  down  the 
layers  of  air  and  would  manifest  itself  by  a  bluish  elec- 
trical discharge  between  the  metal  plates  and  the  glass. 
On  the  other  hand,  the  gradient  of  5880  volts  per  cen- 
timeter would  be  far  below  the  stress  necessary  to 
rupture  the  glass.  Nevertheless  a  discharge  across  air 
spaces  should  always  be  avoided  in  practical  designs 
because  of  its  injurious  effect  on  the  metal  surfaces  and 
also  on  certain  types  of  msulatmg  material.  It  should 
be  observed  that  the  introduction  of  the  glass  plate 
has  appreciably  increased  the  capacity  of  the  con- 
denser. For  example,  with  the  same  voltage  {E  =  10,000) 
as  before,  the  total  flux  is  now  ^  =  AD  =  1000  (8.84  X  io~^^ 
X41, 100)  =3.63X10"^  coulombs.  This  value  is  about 
double  the  value  calculated  with  only  air  between  the 
condenser  plates. 

As  a  practical  application  of  the  principles  governing 
the  behavior  of  condensers  in  series,  consider  the  insu- 
lation between  the  coils  and  core  of  an  air-cooled  trans- 
former, i.e.,  of  which  the  coils  are  not  immersed  in  oil. 
In  addition  assume  the  insulation  to  consist  of  layers  of 
different  materials  made  up  as  follows: 


Total  Thickness. 

Material. 

Mils. 

Centi- 
meters. 

Constant,  k. 

Cotton  braiding  and  varnished  cambric 
Micanite 

70 

125 

62 

24 

0.178 
0.317 
0.158 
0.061 

5 
6 

3 

I 

Air  spaces  (estimated) 

INSULATION  OF  HIGH-PRESSURE  TRANSFORMERS       45 

Then,  suppose  it  is  desired  to  determine  how  high 
an  alternating  voltage  can  be  appKed  between  the  coils 
and  the  core  before  the  maximum  stress  in  the  air 
spaces  exceeds  31,000  volts  per  centimeter,  the  gradient 
which  will  cause  disruption  and  static  discharge,  with 
the  consequent  danger  to  the  insulation  due  to  local 
heating  and  chemical  action.  Assummg  the  coil  to 
constitute  one  flat  plate  of  a  condenser  of  which  the 
other  plate  is  the  iron  frame  or  core,  the  efifect  is  that 
of  a  number  of  plate  condensers  in  series  the  total  elas- 
tance  being 

C    Ci    C2    Cs    C4 

By  Formula   (8),   the  individual  capacities  for  the 

k 
same  surface  area  are  proportional  to  j,  and 

KA_h     h  ,hl4 

C         ki      k2      ks      k^ 

Since 

KA     KAE    KE      KB       E 


C  ^  D         KGatr      Gair 

the  permissible  maximum  value  of  E  is 

/o.iyS    0.317    0.158    o.o6i\ 

=  6260  volts  (maximum). 

The  r.m.s.  value  of  the  corresponding  sinusoidal  alter- 

1  •     6260  ,  .  1     .      ,      ,.    .  . 

natmg  voltage  is   — -^  =  4430,   which  is   the   hmitmg 

V2 


46  PRINCIPLES  OF  TRANSFORMER  DESIGN 

potential  difference  between  windings  and  grounded 
metal  work  if  the  formation  of  corona  is  to  be  avoided. 
A  transformer  having  insulation  made  up  as  previously 
described  would  be  suitable  for  a  6600-volt  three-phase 
circuit  with  grounded  neutral;  but  for  higher  voltages 
the  insulation  should  be  modified,  or  oil  immersion 
should  be  employed  to  fill  all  air  spaces.  If  the  oil- 
cooled  construction  is  employed,  the  previously  con- 
sidered insulations  (slightly  modified  in  view  of  pos- 
sible action  of  the  oil  upon  the  varnish)  would  probably 
be  suitable  for  working  voltages  up  to  15,000. 

13.  Surface  Leakage.  A  large  factor  of  safety  must 
be  allowed  when  determining  the  distance  between 
electrodes  measured  over  the  surface  of  an  insulator. 
Whether  or  not  spark-over  will  occur  depends  not  only 
upon  the  condition  of  the  surface  (clean  or  dirty,  dry 
or  damp),  but  also  upon  the  shape  and  position  of  the 
terminals  or  conductors.  It  is  therefore  almost  impos- 
sible to  determine,  other  than  by  actual  test,  what  will 
happen  in  the  case  of  any  departure  from  standard 
practice.  Surface  leakage  occurs  under  oil  as  well  as 
in  air,  but  generally  speaking,  the  creepage  distance 
under  oil  need  be  only  about  one-quarter  of  what  is 
necessary  in  air. 

An  important  point  to  consider  in  connection  with 
surface  leakage  is  illustrated  by  Figs.  21  and  22.  In 
Fig.  21,  a  thin  disk  of  porcelain  (or  other  solid  insulator) 
separates  the  two  electrodes,  while  in  Fig.  22,  the  same 
material  is  in  the  form  of  a  thick  block  providing  a 
leakage  path  (/)  of  exactly  the  same  length  as  in  Fig.  21. 
The  voltage  required  to  cause  spark-over  will  be  con- 


INSULATION  OF  HIGH-PRESSURE  TRANSFORMERS       47 

siderably  greater  for  the  block  of  Fig.  22  than  for  the 
disk  of  Fig.  21.  This  condition  exists  because  the  flux 
concentration  due  to  the  nearness  of  the  terminals  in 
Fig.  21  begins  breaking  down  the  layers  of  air  around 
the  edges  of  the  electrodes  at  a  much  lower  total  poten- 
tial difference  than  will  be  necessary  in  the  case  of  the 
thicker  block  of  Fig.  22.  The  effect  of  the  incipient 
breakdown  is,  virtually,  to  make  a  conductor  of  the  air 


Fig.  21.  Fig.  22. 

Fig.  21. — Surface  Leakage  over  Thin  Plate. 

Fig.  22. — Surface  Leakage  Over  Thick  Insulating  Block. 


around  the  edges  of  the  metal  electrodes,  and  a  very 
slight  increase  in  the  pressure  will  often  suffice  to  break 
down  further  layers  of  air  and  so  result  in  a  discharge 
over  the  edges  of  the  insulating  disk.,  The  phenomenon 
of  so-called  surface  leakage  may  thus  be  considered 
as  largely  one  of  flux  concentration  or  potential  gra- 
dient. Sometimes  it  will  be  easier  to  eHminate  trouble 
due  to  surface  leakage  by  altering  the  design  of  ter- 


48  PRINCIPLES  OF  TRANSFORMER  DESIGN 

minals  and    increasing  the  thickness  of  the  insulation 
than  by  adding  to  the  length  of  the  creepage  paths. 

14.  Practical  Rules  Applicable  to  the  Insulation  of 
High-voltage  Windings.  For  working  pressures  up  to 
12,000  volts,  solid  insulation,  including  cotton  tape, 
micanite,  pressboard,  horn  paper,  or  any  insulating 
material  of  good  quahty  used  to  separate  the  windmgs 
from  the  core  or  framework,  should  have  a  total  thick- 
ness of  approximately  the  following  values: 


Voltage. 

Thickness  of  Insulation  (Mils) 

no 

40 

400 

45 

1,000 

65 

2,200 

90 

6,600 

180 

12,000 

270 

In  large  high-voltage  power  transformers,  cooled  by 
air  blast,  the  air  spaces  are  rehed  upon  for  insulation. 
The  clearances  between  coils  and  core  or  case  are  neces- 
sarily much  larger  than  in  oil-cooled  transformers,  and 
calculations  similar  to  the  example  previously  worked 
out  should  be  made  to  determine  whether  or  not  the 
insulation  is  sufficient  and  suitably  proportioned  to 
prevent  brush  discharge.  The  calculations  are  made 
on  the  basis  of  several  plate  condensers  in  series;  thus 
the  flux  density  and  dielectric  stress  in  the  various 
layers  of  insulation  can  be  approximately  predetermined. 
The  difficulty  of  avoiding  static  discharges  will  generally 


INSULATION  OF  HIGH-PRESSURE  TRANSFORMERS       49 

stand  in  the  way  of  designing  economical  air-cooled 
transformers  for  pressures  much  in  excess  of  30,000 
volts.    A  rough  rule  for  air  clearance  is  to  allow  a 

distance  equal  to inches,  where  kv   stands  for 

4 
the  virtual  value  of    the    alternating  potential  differ- 
ence in  kilovolts  between  the  two  surfaces  considered. 

With  oil-immersed  transformers,  the  oil  channels 
should  be  at  least  0.25  in.  wide  in  order  that  there  may 
be  free  circulation  of  the  oil.  In  high-voltage  trans- 
formers having  a  considerable  thickness  of  insulation 
between  coils  and  core,  it  is  advantageous  to  divide  the 
oil  spaces  by  partitions  of  pressboard  or  similar  mate- 
rial. Assuming  the  total  thickness  of  oil  to  be  no 
greater  than  that  of  the  soHd  insulation,  a  safe  rule  is 
to  allow  I  mil  for  every  25  volts.  For  instance,  a  total 
thickness  of  insulation  of  i  in.  made  up  of  0.5  in.  of 
sohd  insulation  and  two  0.25  in.  oil  ducts  would  be  suit- 
able for  a  working  pressure  not  exceeding  25X1000  = 
25,000  volts.  Further  particulars  relating  to  oil  insula- 
tion will  be  given  later. 

It  is  customary  to  hmit  the  volts  per  coil  to  5000,  and 
the  volts  between  layers  of  winding  to  400.  Special 
attention  must  be  paid  to  the  insulation  under  the 
finishing  ends  of  the  layers  by  providing  extra  insula- 
tion ranging  from  thin  paper  to  Empire  cloth  or  even 
thin  fuUerboard,  the  material  depending  upon  the  voltage 
and  also  upon  the  amount  of  mechanical  protection 
required  to  prevent  cutting  through  the  insulation  where 
the  wires  cross.  Sometimes  the  insulation  is  bent 
around  the  end  wires  of  a  layer  to  prevent  breakdown 


50  PRINCIPLES  OF  TRANSFORMER  DESIGN 

over  the  ends  of  the  coil.  Where  space  permits,  however, 
the  layers  of  insulation  may  be  carried  beyond  the  ends  of 
the  winding  so  as  to  avoid  surface  leakage.  This  arrange- 
ment is  more  easily  carried  out  in  core-t^-pe  transformers 
than  in  shell-type  units.  A  practical  rule  for  deter- 
mining the  surface  distance  (in  inches)  required  to  pre- 
vent leakage  (given  by  Messrs.  Fleming  and  Johnson 
in  the  book  previously  referred  to)  is  to  allow  0.5  in. 
-I-0.5  X  kilovolts,  when  the  surfaces  are  in  air.  For  sur- 
faces under  oil,  the  allowance  may  be  0.5+0.1  Xkilovolts. 
In  any  case  it  is  important  to  see  that  the  creepage  sur- 
faces are  protected  as  far  as  possible  from  deposits  of 
dirt.  When  the  coils  of  shell-type  transformer  are 
''sandwiched,"  it  is  customary  to  use  half  the  normal 
number  of  turns  in  the  low-tension  coils  at  each  end  of 
the  stack.  This  has  the  advantage  of  keeping  the  high- 
tension  coils  well  away  from  the  iron  stampings  and 
clamping  plates  or  frame. 

Extra  Insulation  on  End  Turns.  Concentration  of 
potential  between  turns  at  the  ends  of  the  high-tension 
winding  is  liable  to  occur  with  any  sudden  change  of 
voltage  across  the  transformer  terminals,  such  as  when 
the  supply  is  switched  on,  or  when  lightning  causes 
potential  disturbances  on  the  transmission  lines.  It  is, 
therefore,  customary  to  pay  special  attention  to  the  insu- 
lation of  the  end  turns  of  the  high-tension  winding. 
Transformers  for  use  on  high-voltage  circuits  usually 
have  about  75  ft.  at  each  end  of  the  high-tension  winding 
insulated  to  withstand  three  to  four  times  the  voltage 
between  turns  that  would  puncture  the  insulation  in  the 
body  of  the  winding. 


INSULATION  OF  HIGH-PRESSURE  TRANSFORMERS       51 

It  is  very  difficult  to  predetermine  the  extra  pressure 
to  which  the  end  turns  of  a  power  transformer  con- 
nected to  an  overhead  transmission  line  may  at  times 
be  subjected,  but  it  is  safe  to  say  that  the  instantaneous 
potential  difference  between  turns  may  occasionally  be 
of  the  order  of  forty  to  fifty  times  the  normal  working 
pressure.  In  such  cases  the  usual  strengthening  of  the 
insulation  on  the  end  turns  would  not  afford  adequate 
protection,  and  for  this  reason  a  separate  specially 
designed  reactance  coil  connected  to  each  end  of  the  high- 
tension  winding  would  seem  to  be  the  best  means  of 
guarding  against  the  effects  of  surges  or  sudden  changes 
of  pressure  occurring  in  the  electric  circuit  outside  the 
transformer.  The  theory  of  abnormal  pressure  rises  in 
the  end  sections  of  transformer  windings  will  not  be 
discussed  here. 

15.  Winding  Space  Factor.  Kjiowing  the  thickness 
of  the  cotton  covering  on  the  wires,  the  insulation 
"between  layers  of  winding,  between  coil  and  coil 
and  between  coil  and  iron  stampings,  it  becomes 
an  easy  matter  to  determine  approximately  the 
total  cross-section  of  the  winding-space  to  accom- 
modate  a   given   cross-section  of   copper.     The    ratio 

cross-section  of  copper  ,  •  ,    •     ,  , 

7^-^ ,  which  IS  known  as  the 

cross-section  of  winding  space 

space  factor,  will  naturally  decrease  with  the  higher 

voltages  and  smaller  sizes  of  wire.     This  factor  may  be 

as  high  as  0.46  in  large  transformers  for  pressures  not 

exceeding  2200  volts;    in  33, 000-volt  transformers  for 

outputs  of  200  k.v.a.  and  upward  it  will  have  a  value 

ranging  between  0.35  and  0.2,  while  in  oil-immersed 


52  PRINCIPLES  OF  TRANSFORMER  DESIGN 

power  transformers  for  use  on  ioo,ooo-volt  circuits  the 
factor  may  be  as  low  as  0.06. 

16.  Oil  Insulation.  There  is  a  considerable  amount  of 
published  matter  relating  to  the  properties  of  insulating 
oils,  and  also  to  the  various  methods  of  testing,  puri- 
fying, and  drying  oils  for  use  in  transformers.  A  con- 
cise statement  of  the  points  interesting  to  those  installing 
or  having  charge  of  transformers  will  be  found  in  W.  T. 
Taylor's  book  on  transformers.*  What  follows  here  is 
intended  merely  as  a  guide  to  the  designer  in  providing 
the  necessary  clearances  to  avoid  spark-over,  including  a 
reasonable  factor  of  safety. 

Mineral  oil  is  generally  employed  for  insulating  pur- 
poses, its  main  function  in  transformers  being  to  trans- 
fer the  heat  by  convection  from  the  hot  surfaces  to  the 
outside  walls  of  the  containing  case,  or  to  the  cooling 
coils  when  these  are  provided.  The  presence  of  an 
extremely  small  percentage  of  water  reduces  the  insu- 
lating properties  of  oil  considerably.  It  is  therefore 
important  to  test  transformer  oil  before  usmg  it,  and  if 
necessary  extract  the  moisture  by  filtering  through  dry 
blotting  paper,  or  by  any  other  approved  method.  Dry 
oil  will  withstand  pressures  up  to  50,000  volts  (alter- 
nating) between  brass  disks  0.5  in.  in  diameter  with  a 
separation  of  0.2  in.  For  use  in  high-voltage  trans- 
formers, the  oil  should  be  required  to  withstand  a  test 

*  "  Transformer  Practice,"  by  W.  T.  Taylor— McGraw-Hill  Book 
Company,  Inc.  For  further  information  refer  H.  W.  Tobey  on  the 
"Dielectric  Strength  of  Oil"— Trans.  A.I.E.E.;  Vol.  XXIX,  page 
1 189  (1910).  Also  "  Insulating  Oils,"  Jotirn.  Inst.  E.E.,  Vol.  54,  page 
497  (1916). 


INSULATION  OF  HIGH-PRESSURE  TRANSFORMERS       53 

of  45,000  volts  under  the  above  conditions.  The  good 
insulating  qualities  of  oil  suggest  that  only  small  clear- 
ances would  be  required  in  transformers,  even  for  high 
voltages;  but  the  form  of  the  surfaces  separated  by  the 
layer  of  oil  will  have  a  considerable  effect  upon  the  con- 
centration of  flux  density,  and  therefore  upon  the  volt- 
age gradient.  As  an  example,  if  100,000  volts  breaks 
down  a  i-in.  layer  of  a  certain  oil  between  two  parallel 
disks  4  in.  in  diameter,  the  same  pressure  will  spark 
across  a  distance  of  about  3.5  in.  between  a  disk  and  a 
needle  point. 

Partitions  of  soHd  insulation  such  as  pressboard  or 
fullerboard  are  always  advisable  in  the  spaces  occu- 
pied by  the  oil,  since  they  will  prevent  the  lining  up  of 
partly  conducting  impurities  along  the  lines  of  force 
and  reduce  the  total  clearance  which  would  otherwise 
be  necessary. 

In  a  transformer  oil  of  average  quality,  the  sparking 
distance  between  a  needle  point  and  a  flat  plate  is  approx- 
imately (0.25 +0.04 Xkv.)  inches.  Since  there  may 
be  sharp  corners  or  irregularities  corresponding  to  a 
needle  point,  which  will  produce  concentration  of 
dielectric  flux,  it  therefore  seems  advisable  to  introduce 
a  factor  of  safety  for  oil  spaces  between  high  tension 
and  grounded  metal — for  instance,  between  the  ends  of 
high-tension  coils  and  the  containing  case — by  basing 
the  oil  space  dimension  on  the  formula, 

Thickness  of  ofl  (inches)  =0.25-1-0.1  Xkv.,    .     (11) 
where  kv.  stands  for  the  working  pressure  in  kilovolts. 


54  PRINCIPLES  OF  TRANSFORMER  DESIGN 

With  one  or  two  partitions  of  solid  insulating  mate- 
rial dividing  the  oil  space  into  sections,  the  total  thick- 
ness need  not  exceed 

0.25 +0.065  Xkv (12) 

If  the  total  thickness  of  solid  insulation  is  about  equal 
to  that  of  the  oil  ducts  (not  an  unusual  arrangement 
between  coils  and  core),  the  rule  previously  given  for 
solid  insulation  may  be  slightly  modified  to  include  a 
minimum  thickness  of  0.25  in.,  and  put  in  the  form, 

Total  thickness  of  oil  ducts  plus  1 

solid     insulation     of     approxi-  >  =0.25+0.03 Xkv.  (13) 

mately  equal  thickness  (inches)  J 

A  suitable  allowance  for  surface  leakage  under  oil,  in 
inches,  as  already  given,  is 

0.5+0.1  Xkv (14) 

17.  Tenninals  and  Bushings.  The  exact  pressure 
which  will  cause  the  breakdown  of  a  transformer  ter- 
minal bushing  generally  has  to  be  determined  by  test, 
because  the  shape  and  proportions  of  the  metal  parts 
are  rarely  such  that  the  concentration  of  flux  density 
at  corners  or  edges  can  be  accurately  predetermined.* 

*  The  reader  who  desires  to  go  deeply  into  the  study  of  high-pressure 
terminal  design  should  refer  to  the  paper  by  Mr.  Chester  W.  Price 
entitled  "  An  Experimental  Method  of  Obtaining  the  Solution  of  Elec- 
trostatic Problems,  with  Notes  on  High-voltage  Bushing  Design."  Trans. 
A.I.E.E.,  Vol.  36,  page  905  (Nov.,  191 7). 


INSULATION  OF  HIGH-PRESSURE  TRANSFORMERS       55 

However,  there  are  certain  important  points  to  bear 
in  mind  when  designing  the  insulation  of  transformer 
terminals,  and  these  will  now  be  referred  to  briefly. 

The  high-tension  leads  of  a  transformer  may  break 
down  (i)  by  puncture  of  the  insulation,  or  (2)  by 
spark-over  from  terminal  to  case.  If  the  transformer 
lead  could  be  considered  as  an  insulated  cable  with  a 
suitable  dielectric  separating  it  from  an  outer  concentric 


Fig.  23. — Section  through  Insulated  Conductor. 

metal  tube  of  considerable  length,  the  calculation  of 
the  puncture  voltage  (i)  would  be  a  simple  matter. 
For  instance,  let  r  in  Fig.  23  be  the  radius  of  the  inner 
(cylindrical)  conductor,  and  R  the  internal  radius  of 
the  enclosing  tube,  the  space  between  being  filled  with 
a  dielectric  of  which  the  specific  inductive  capacity 
(^)  is  constant  throughout  the  insulating  material. 
The  equipotential  surfaces  will  be  cyHnders,  and  the 


56  PRINCIPLES  OF  TRANSFORMER  DESIGN 

flux  density  over  the  surface  of  any  cylinder  of  radius 
X  and  of  length  i  cm.,  will  be  Z>  =  — • 
By  Formula  (lo)  the  potential  gradient  is, 

D         ^  ,    X 

^~Kk~2irxKk ^^^^ 


In  order  to  express  this  relation  in  terms  of  the  total 
voltage  E,  it  is  necessary  to  substitute  for  the  symbol 
-ir  its  equivalent  ExC,  and  calculate  the  capacity 
C  of  the  condenser  formed  by  the  rod  and  the  con- 
centric tube.  Considering  a  number  of  concentric  shells 
in  series,  the  elastance  may  be  written  as  follows: 

I      f^    dx  1     .       R  ,   .. 

Substituting  in  (15),  we  have, 

G  = ^  volts  per  centimeter,    .     .     (17) 

X  loge  — 


the  maximum  value  of  which  is  at  the  surface  of  the 
inner  conductor,  where 

E 


1       ^' 

r  log.  - 


(18) 


This  formula  is  of  some  value  in  determining  the  thick- 
ness of  insulation  necessary  to  avoid  overstressing  the 


INSULATION  OF  HIGH-PRESSURE  TRANSFORMERS        57 

dielectric;  but  it  is  not  strictly  applicable  to  trans- 
former bushings  in  which  the  outer  metal  surface  (the 
bushing  in  the  hd  of  the  containing  tank)  is  short  in 
comparison  with  the  diameter  of  the  opening.  The 
advantage  of  havmg  a  fairly  large  value  for  r  is  indicated 
by  Formula  (i8),  and  a  good  arrangement  is  to  use  a 
hollow  tube  for  the  high-tension  terminal,  with  the  lead 
from  the  windings  passing  up  through  it  to  a  clamping 
terminal  at  the  top. 

Sohd  porcelain  bushings  with  either  smooth  or  cor- 
rugated surfaces  may  be  used  for  any  pressure  up  to 
40,000  volts,  but  for  higher  pressures  the  oil-filled  type 
or  the  "  condenser  "  type  of  terminal  is  preferable.  In 
designmg  plain  porcelain  bushings  it  is  important  to 
see  that  the  potential  gradient  in  the  air  space  between 
the  metal  rod  and  the  insulator  is  not  hable  to  cause 
brush  discharge,  as  this  would  lead  to  chemical  action, 
and  a  green  deposit  of  copper  nitrate  upon  the  rod.  The 
calculations  would  be  made  as  explained  for  the  parallel- 
plate  condensers  in  which  a  sheet  of  glass  was  inserted 
(see  "  Capacities  in  Series  "),  except  that  the  elastances 
of  the  condensers  are  now  expressed  by  Formula  (16). 

18.  Oil-filled  Bushings.  The  chief  advantages  of  a 
hollow  insulating  shell  filled  with  oil  or  insulating  com- 
pound that  can  be  poured  in  the  Uquid  state,  are  the 
absence  of  air  spaces  where  corona  may  occur,  and  the 
possibiHty  of  obtaining  a  more  uniform  and  rehable 
insulation  than  with  sohd  insulators— such  as  porcelain, 
when  the  thickness  is  considerable.  The  metal  ring 
by  which  such  an  insulator  (see  Fig.  24)  is  secured  to  the 
transformer  cover  usually  takes  the  form  of  a  cyhnder 


68  PRINCIPLES  OF  TRANSFORMER  DESIGN 

of  sufficient  length  to  terminate  below  the  surface  of  the 
oil.  The  advantage  of  this  arrangement  is  that  the 
dielectric  flux  over  the  surface  of  the  lower  part  of  the 
insulator  is  through  oil  only,  and  not  as  would  otherwise 
be  the  case,  through  oil  and  air.  With  the  two  mate- 
rials of  different  dielectric  constants,  the  stress  at  the 
surface  of  the  oil  may  exceed  the  dielectric  strength  of 
air,  in  which  case  there  would  be  corona  or  brush  dis- 
charge which  might  practically  short-circuit  the  air 
path  and  increase  the  stress  over  that  portion  of  the 
surface  which  is  under  the  oil. 

The  bushing  illustrated  in  Fig.  24  has  been  designed 
for  a  working  pressure  of  88,000  volts  between  high- 
tension  terminal  and  case,  the  method  of  computation 
being,  briefly,  as  follows:  Applying  the  rule  for  sur- 
face leakage  distances  previously  given,  this  dimension 
is  found  to  be  0.5+-®^  =  44.5  in.  The  insulator  need 
not,  however,  measure  44.5  in.  in  height  above  the 
cover  of  the  transformer  case,  because  corrugations  can 
be  used  to  obtain  the  required  length.  A  safe  rule  to 
follow  in  deciding  upon  a  minimum  height,  i.e.,  the 
direct  distance  in  air  between  the  terminal  and  the 
grounded  metal,  is  to  make  this  dimension  at  least  as 
great  as  the  distance  between  needle  points  that  would 
just  withstand  the  test  voltage  without  sparking  over. 
The  test  pressure  is  usually  twice  the  working  pressure 
plus  1000  volts,  or  177  kv.  (r.m.s.  value)  in  this  par- 
ticular case.  This  value  corresponds  to  a  distance  of 
about  48  cm.,  or  (say)  19  in.  In  order  that  there  may 
be  an  ample  margin  of  safety,  it  will  be  advisable  to 
make  the  total  height  of  the  insulator  not  less  tlian  22 


INSULATION  OF  HIGH-PRESSURE  TRANSFORMERS       59 


JHetel  tube  of 
Si  outside  diameter 


Top  of 

Transformei! 


'mm 


Iron  Sleeve  carried 
'below  surface  ol  oU 


|.^In3ulatlng  tube  around 
if  metal  cap  and  transformer 


H.T.  Lead 


Fig.  24. — Three-part  Composition-filled  Porcelain  Transformer  Bushing, 
Suitable  for  a  Working  Pressure  of  88,000  Volts  to  Ground. 


60  PRLN'CIPLES  OF  TRANSFORMER  DESIGN 

in.,  apart  from  the  number  or  depth  of  the  corrugations. 
The  actual  height  in  Fig.  24  is  31  in.  because  the  cor- 
rugations on  the  outside  of  the  porcelain  shell  are  neither 
very  numerous  nor  very  deep.  In  this  connection  it 
may  be  stated  that  a  short  insulator  with  deep  corruga- 
tions designed  to  provide  ample  surface  distance  is  not 
usually  so  effective  as  a  tall  insulator  with  either  a 
smooth  surface  or  shallow  corrugations.  The  reason 
is  that  much  of  the  dielectric  flux  from  the  high-tension 
terminal  to  the  external  sleeve  or  supporting  framework 
passes  through  the  flanges,  the  specific  inductive  capacity 
of  which  is  two  to  three  times  that  of  the  air  between 
them.  The  result  is  an  increased  stress  in  the  air  spaces, 
which  is  equivalent  to  a  reduction  in  the  effective  height 
of  the  insulator. 

In  the  design  under  consideration  it  is  assumed  that 
the  hollow  (porcelain)  shell  is  filled  with  an  insulating 
compound  which  is  solid  at  normal  temperatures,  and 
that  the  joints  therefore  need  not  be  so  carefully  made 
as  when  oil  is  used.  The  insulator  consists  of  three 
parts  only,  which  are  jointed  as  indicated  on  the  sketch. 
Oil-filled  bushings  for  indoor  use  generally  have  a  large 
number  of  parts,  usually  in  the  form  of  flanged  rings 
with  molded  tongue-and-groove  joints  filled  with  a 
suitable  cement.  There  is  always  the  danger,  however, 
that  a  vessel  so  constructed  may  not  be  quite  oil-tight, 
therefore  the  solid  compound  has  an  advantage  over  the 
oil  in  this  respect. 

The  creepage  distance  over  the  surface  of  the  insulator 
in  oil  may  be  very  much  less  than  in  air.  Applying  the 
rule  previously  given,   the  minimum  distance  in  this 


INSULATION  OF  HIGH-PRESSURE  TRANSFORMERS       61 

case  would  be  o.54-(o.i  X88)  =9.3  in.  In  the  design 
illustrated  by  Fig.  24,  however,  this  dimension  has  been 
increased  about  50  per  cent  with  a  view  to  keeping 
the  high-tension  connections  well  away  from  the  sur- 
face of  the  oil  and  grounded  metal.  To  prevent  the 
accumulation  of  conducting  particles  in  the  oil  along  the 
lines  of  stress,  and  afford  increased  protection  with  only 
a  small  addition  in  cost,  it  is  advisable  to  slip  one  or 
more  insulating  tubes  over  the  lower  part  of  the  ter- 
minal, as  indicated  by  the  dotted  Hues  in  the  sketch. 
Corrugations  on  the  surface  of  the  insulator  in  the  oil 
are  usually  unnecessary,  and  sometimes  objectionable 
because  they  collect  dirt  which  may  reduce  the  effective 
creepage  distance. 

Having  decided  upon  the  height  and  surface  distances 
to  avoid  all  danger  of  spark-over,  the  problem  which 
remains  to  be  dealt  with  is  the  provision  of  a  proper  t]iick- 
ness  of  insulation  to  prevent  puncture.  In  order  to 
avoid  complication  of  the  problem  by  considering  the 
different  dielectric  constants  {k)  of  the  compound  used 
for  filling  and  of  the  external  shell  (assumed  in  this 
case  to  be  porcelain),  it  may  be  assumed  either  that 
there  is  no  difference  in  the  dielectric  constants  of  the 
two  materials,  or  that  the  thickness  of  the  inclosing 
shell  of  porcelain  is  neghgibly  small  in  relation  to  the 
total  external  diameter  of  the  insulator.  Either  assump- 
tion, neglecting  the  error  due  to  the  Hmited  length  of  the 
external  metal  sleeve,*  permits  the  use  of  Formula  (18), 

*  The  maximum  stress  in  the  dielectric  might  be  5  to  10  per  cent 
greater  than  calculated  by  using  formulas  relating  to  very  long  cylin- 
ders. The  corners  at  the  ends  of  the  outer  cylinder  should  be  rounded 
off  to  avoid  concentration  of  dielectric  flux  at  these  places. 


62  PRINCIPLES  OF  TRANSFORMER  DESIGN 

giving  the  relation  between  the  maximum  potential 
gradient  and  the  dimensions  of  the  bushing,  without 
correction. 

Suppose  that  the  disruptive  gradient  of  the  insulating 
compound  is  90  kv.  per  centimeter  (maximum  value)  or 
63.5  kv.  per  centimeter  (r.m.s.  value)  of  the  alternating 
voltage.  With  a  test  pressure  of  177  kv.  and  a  margin  of 
safety  of  25  per  cent,  the  value  of_£  in  Formula  (18) 
will  therefore  be  £  =  177X1.25X^2=313  kv. 

Since  the  disadvantage  of  a  very  small  value  of  r  is 

evident  from  an  inspection  of  the  formula,  the  outside 

diameter  of  the  inner  tube  is  made  2.25  in.     Then,  since 

77 
G=- 


1      ^ 

r  log.  - 


Wio-  = ^^ =  1.216, 

r     2. 54X1. 125X90X2.303 

whence  7^  =  3.79,  or  (say)  3.75  in.  An  external  diam- 
eter of  7.5  in.  at  the  center  of  the  insulator  will  there- 
fore be  sufficient  to  prevent  the  stress  at  any  point 
exceeding  the  rupturing  value  even  under  the  test  pres- 
sure. 

19.  Condenser  Type  of  Bushing.  If  the  total  thick- 
ness of  the  insulation  between  the  high-tension  rod  and 
the  (grounded)  supporting  sleeve  is  divided  into  a  num- 
ber of  concentric  layers  by  metallic  cylinders,  the  con- 
centration of  dielectric  flux  at  certain  points  (leading  to 
high  values  of  the  voltage  gradient)  is  avoided.  The 
bushing  then  consists  of  a  number  of  plate  condensers 
in  series,  with  a  definite  potential  difference  between 


INSULATION  OF  HIGH-PRESSURE  TRANSFORMERS        63 

the  plates.  If  the  total  radial  depth  of  insulation  is 
divided  into  a  large  number  of  concentric  layers  (of 
the  same  thickness),  separated  by  cylinders  of  tinfoil 
(of  the  same  area),  the  several  condensers  would  all  have 
the  same  capacity.  The  dielectric  flux  density,  and 
therefore  the  potential  gradient,  would  then  be  the  same 
in  all  the  condensers,  so  that  the  outer  layers  of  insula- 
tion would  be  stressed  to  the  same  extent  as  the  inner 
layers,  and  the  total  radial  depth  of  insulation  would 
be  less  than  when  the  stress  distribution  follows  the 
logarithmic  law  (Formula  i8)  as  in  the  case  of  the  soHd 
porcelain,  or  oil-filled,  bushing. 

The  section  on  the  right-hand  side  of  Fig.  25  is  a 
diagrammatic  representation  of  a  condenser  bushing 
shaped  to  comply  with  the  assumed  conditions  of  equal 
thicknesses  of  insulation  and  equal  areas  of  the  con- 
denser plates.  With  a  sufficient  number  of  concentric 
layers,  the  condition  of  equal  potential  difference  be- 
tween plate  and  plate  throughout  the  entire  thickness 
would  be  approximated;  but  the  creepage  distance  over 
the  insulation  between  the  edges  of  the  metal  cylinders 
would  be  much  smaller  for  the  outer  layers  than  for 
layers  nearer  to  the  central  rod  or  tube.  It  is  equally, 
if  not  more,  important  to  prevent  excessive  stress  over 
the  surface  than  in  the  body  of  the  insulator,  and  a 
practical  condenser  type  of  terminal  can  be  designed 
as  a  compromise  between  the  two  conflicting  require- 
ments. By  making  the  terminal  conical  in  form,  as 
indicated  by  the  dotted  lines  on  the  right-hand  side 
and  the  full  lines  on  the  left-hand  side  of  the  sketch 
(Fig.  25),  neither  of  the  ideal  conditions  will  be  exactly 


64 


PRINCIPLES  OF  TRANSFORMER  DESIGN 


fulfilled,  but  practical  terminals  so  constructed  are  easily 
manufactured,  and  give  satisfaction  on  circuits  up  to 


Metal  pliicld  to  coiurol 
^distribution  of  dielectric  field 


V. 


100,000  Volts 
807o~Oo~VoTti"" 
607ooo~vbTts 


40,000  VoHs 


20,000 


^//M///M///^//,' 


Zero  poteatial 


ij^infoll 

\  i/Insulation 


/ 


^Grounded 
metal 


Metal  tube  or  rod, 
forming  H.T.  lead, 

Fig.  25. — Illustrating  Principle  of  Condenser  Type  Bushing. 

150,000  volts.  By  varying  the  thickness  of  the  indi- 
vidual insulating  cylinders,  it  is  an  easy  matter  to 
design  a  condenser  type  terminal  of  which  the  con- 


INSULATION  OF  HIGH-PRESSURE  TRANSFORMERS        65 

densers  in  series  all  have  the  same  capacity  even  while 
the  outside  surface  is  conical  in  shape  as  shown  on  the 
left-hand  side  of  Fig.  25.  This  gives  a  uniform  potential 
gradient  along  the  surface,  and  results  in  a  good  practical 
form  of  condenser-t^pe  bushing. 

If  the  ends  of  the  metal  cyHnders  coincide  with  equi- 
potential  surfaces  having  the  same  potential  as  that 
which  they  themselves  attain  by  virtue  of  the  respective 
capacities  of  the  condensers  in  series,  there  will  be  no 
corona  or  brush  discharge  at  the  edges  of  these  cylin- 
ders. This  ideal  condition  is  represented  diagram- 
matically  in  Fig.  25,  where  a  large  metal  disk  is  shown 
at  the  top  of  the  terminal.  The  object  of  this  metal 
shield  is  to  distribute  the  field  between  the  terminal 
and  the  transformer  cover  in  such  a  manner  as  to  satfsfy 
the  above-mentioned  condition.  In  practice,  the  ten- 
dency for  corona  to  form  at  the  exposed  ends  of  the  tin- 
foil cylinders  is  counteracted  by  treating  the  finished 
terminal  with  several  coats  of  varnish,  and  surrounding 
it  with  an  insulating  cylinder  filled  with  an  insulating 
compound  which  can  be  poured  in  the  Hquid  form  and 
which  solidifies  at  ordinary  temperatures.  This  con- 
struction is  shown  in  Fig.  26,  which  represents  a  prac- 
tical terminal  of  the  condenser  type.  Compared  with 
Fig.  24,  it  is  longer,  but  appreciably  smaller  in  diameter 
where  it  passes  through  the  transformer  cover. 

The  dimensions  of  a  condenser-type  terminal  such 
as  illustrated  in  Fig.  26  may  be  determined  approxi- 
mately as  follows:  Assuming  the  working  pressure  as 
88,000  volts,  and  the  maximum  permissible  potential 
gradient  in  the  dielectric  (usually  consisting  of  tightly 


66  PRINCIPLES  OF  TRANSFORMER  DESIGN 


Fig.  26. — Condenser-type  Transformer  Bushing  Suitable  for  a  Working 
Pressure  of  88,000  Volts. 


INSULATION  OF  HIGH-PRESSURE  TRANSFORMERS        67 

wound   layers   of   specially  treated  paper)  as  90  kv.,* 
the  maximum  radial  thickness  of  insulation  required 

will  be        total  volts         W^   ^^g  cm.   or   (say)    1.5 

voltage  gradient  90 
in.  to  include  an  ample  allowance  for  the  dividing 
layers  of  metal  foil.  If  the  inner  tube  is  2.25  in.  in 
diameter,  as  in  the  previous  example,  the  external 
diameter  over  the  insulation  at  the  center  will  be  2.25 
X3  =  5.25  in.  instead  of  the  7.5  in.  required  for  the 
previous  design. 

It  is  customary  to  allow  about  4000  volts  per  layer, 
and  twenty-two  layers  of  insulation  alternating  with 
twenty-two  layers  of  tinfoil  are  used  in  this  particular 
design.  It  is  true  that  ideal  conditions  will  not  be 
actually  fulfilled;  the  aggregate  thickness  of  insulation 
might  have  to  be  slightly  greater  than  1.5  in.,  but  the 
inner  tube  might  be  made  1.75  in.  or  2  in.  instead  of 
2.25  in.,  and  a  practical  terminal  for  88, 000- volt  service 
could  undoubtedly  be  constructed  with  a  diameter  over 
the  insulation  not  exceeding  5.25  in. 

The  projection  of  the  terminal  above  the  grounded 
plate  (the  cover  of  the  transformer  case)  need  not  be 
so  great  as  would  be  indicated  by  the  appHcation  of  the 
practical  rule  previously  given  for  surface  leakage  dis- 
tance, namely,  that  this  distance  should  be  (0.5-! j 

in.,  where  kv.  stands  for  the  working  pressure.  The 
reason  why  a  somewhat  shorter  distance  is  permissible 
is  that  the  surface  of  the  terminal  proper  has  been  cov- 
ered by  varnish  and  a  soHd  compound,  and  so  far  as  the 
enclosing  cylinder  is  concerned,  the  stress  along  the  sur- 
*  Same  as  in  the  example  of  the  compound-filled  insulator. 


68  PRINCIPLES  OF  TRANSFORMER  DESIGN 

face  of  this  cylinder  will  be  fairly  uniform,  especially  if 
a  large  flux-control  shield  is  provided,  as  shown  in  Fig. 
26.  In  order  to  avoid  the  formation  of  corona  at  the 
lower  terminal  (below  the  surface  of  the  oil)  this  end  may 
conveniently  be  in  the  form  of  a  sphere,  the  diameter 
of  which  would  depend  upon  the  voltage  and  the  prox- 
imity of  grounded  metal. 

The  following  particulars  relate  to  a  condenser  type 
bushing  actually  in  service  on  80,000  volts.  The  layers 
of  insulation  are  built  up  on  a  metal  tube  of  2.25  in. 
outside  diameter.  The  diameter  over  the  outside  in- 
sulating cy Under  is  5.3  in.  This  bushing  has  uniform 
capacity,  the  thickness  of  the  inner  and  outer  insulating 
wall  being  the  same,  namely  0.062  in.;  but  the  thickness 
of  the  intermediate  cylinders  is  variable,  the  maximum 
being  0.073  '^^-  ^or  the  twelfth  and  thirteenth  cylinders. 
(A  plot  of  the  individual  thickness  forms  a  hyperboHc 
curve.)  The  static  shield  or  ''  hat  "  is  9  in.  diameter  and 
2  in.  thick,  the  edge  being  rolled  to  a  true  semicircle. 
When  provided  with  a  casing  filled  with  gum,  and  when 
the  taper  is  such  that  the  steps  on  the  air  end  are  1.69  in. 
(total  length  =  1.69X22  =37.2  in.),  there  is  no  dif- 
ficulty in  raising  the  voltage  to  300,000  (r.m.s.  value) 
without  arc-over.  The  same  bushing  without  a  casing 
would  arc-over  at  about  285,000  volts;  but  this  can  be 
raised  to  the  same  value  as  for  the  terminal  with  gum- 
filled  casing  if  the  size  of  the  static  shield  is  increased  to 
about  2  ft.  diameter.  When  the  arc-over  voltage  is 
reached,  the  discharge  takes  place  between  the  edge  of 
the  static  shield  and  the  flange  which  is  bolted  to  the 
transformer  case. 


CHAPTER  III 

EFFICIENCY  AND  HEATING  OF  TRANSFORMERS 

20.  Losses  in  Core  and  Windings.  The  power  loss 
in  the  iron  of  the  magnetic  circuit  is  due  partly  to 
hysteresis  and  partly  to  eddy  currents.  The  loss  due 
to  hysteresis  is  given  approximately  by  the  formula 

Watts  per  pound  =  KnB^J, 

where  Kh  is  the  hysteresis  constant  which  depends  upon 
the  magnetic  qualities  of  the  iron.  The  symbols  B  and/ 
stand,  respectively,  for  the  maximum  value  of  the  mag- 
netic flux  density,  and  the  frequency.  An  approximate 
expression  for  the  loss  due  to  eddy  currents  is 

Watts  per  pound  =  Ke  (Bft)^, 

where  /  is  the  thickness  of  the  laminations,  and  Ke  is  a 
constant  which  is  proportional  to  the  electric  conduc- 
tivity of  the  iron. 

With  the  aid  of  such  formulas,  the  hysteresis  and  eddy 
current  losses  may  be  calculated  separately,  and  then 
added  together  to  give  the  total  watts  lost  per  pound 
of  the  core  material;  but  it  is  more  convenient  to  use 
curves  such  as  those  of  Fig.  27,  which  should  be  plotted 
69 


70 


PRINCIPLES  OF  TRANSFORMER  DESIGN 


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EFFICIENCY  AND  HEATING  OF  TRANSFORMERS      71, 

from  tests  made  on  samples  of  the  iron  used  in  the  con- 
struction of  the  transformer.  These  curves  give  the 
relation  between  maximum  value  of  flux  density,  and 
total  iron  loss  per  pound  at  various  frequencies.  The 
curves  of  Fig.  27  are  based  on  average  values  obtained 
with  good  samples  of  commercial  transformer  iron  and 
silicon-steel;  the  thickness  of  the  laminations  being 
about  0.014  in. 

The  cost  of  silicon-steel  stampmgs  is  greater  than  that 
of  ordinary  transformer  iron;  but  the  smaller  total  iron 
loss  resulting  from  the  use  of  the  former  material  will 
almost  invariably  lead  to  its  adoption  on  economic 
grounds.  The  eddy-current  losses  are  smaller  in  the 
alloyed  material  than  in  iron  laminations  of  the  same 
thickness  because  of  the  higher  electrical  resistance  of 
the  former.  The  permeability  of  silicon-steel  is  shghtly 
lower  than  that  of  ordinary  iron,  and  this  may  lead  to  a 
somewhat  larger  magnetizmg  current;  on  the  other  hand, 
the  modern  alloyed  transformer  material  (silicon-steel)  is 
non-ageing,  that  is  to  say,  it  has  not  the  disadvantage 
common  to  transformers  constructed  fifteen  to  twenty 
years  ago,  in  which  the  iron  losses  increased  appreciably 
during  the  first  two  or  three  years  of  operation.  The 
"^ag|^  "  of  the  ordinary  brands  of  transformer  iron — 
resulting  in  larger  losses — is  caused  by  the  material  being 
maintained  at  a  fairly  high  temperature  for  a  consider- 
able length  of  time. 

The  maximum  flux  density  in  transformer  cores  is 
generally  kept  below  the  knee  of  the  B-H  curve.  As  a 
guide  for  use  in  preliminary  designs,  usual  values  of  B 
(gausses)  are  given  below : 


/t^         principles  of  transformer  design 
Approximate  Values  of  B  in  Transformer  Cores 


/  =  2S 

/  =  S0  or  60 

Small  lighting  or  distributing 

transformers: 

Ordinary  iron 

8,000  to  11,000 

5,000  to    7,000 

Alloyed  iron 

11,000  to  13,000 

9,000  to  11,000 

Power  transformers: 

10,000  to  13,000 
12,000  to  14,000 

9,000  to  11,000 

Alloyed  iron 

11,000  to  14,000 

The  losses  in  the  iron  core  are  usually  less  than  one 
watt  per  pound,  although  they  sometimes  amount  to 
1.5  watts,  and  even  1.8  watts,  per  pound.  The  higher 
figures  apply  to  large,  artificially  cooled,  power  trans- 
formers. 
5^  Current  Density  in  Windings.  Even  with  well-ven- 
tilated coils  (air  blast),  or  improved  methods  of  pro- 
ducing good  oil  circulation,  the  permissible  current  den- 
sity in  the  copper  windings  is  limited  by  local  heatmg. 
If  the  watts  lost  per  pound  of  copper  exceed  a  certain 
amount,  there  will  be  danger  of  internal  temperatures 
sufficiently  high  to  cause  injury  to  the  insulation.  As  a 
r"ough  guide  in  deciding  upon  suitable  values  for  trial 
dimensions  in  a  preUminary  design,  the  following  approx- 
imate figures  may  be  used : 

Average  Values  of  Current  Density  (a)  est 
Commercial  Transformers 

Type  of  Transformer  ^qTa^lTh' 

Standard  lighting  transformers  (oil-immersed;  sel f -cooled) .  800  to  1300 
Transformers  for  use  in  Central  Generating  Stations,  or 

Substations  (oil-cooled,  or  air  blast) iioo  to  1600 

Large,  carefully  designed  transformers,  oil-insulated,  with 

forced  circulation  of  oil,  or  with  water  cooling-coils 1400  to  2000 


EFFICIENCY  AND  HEATING  OF  TRANSFORMERS      73 

When  the  current  is  very  large,  it  is  important  to  sub- 
divide the  conductors  to  prevent  excessive  loss  by  eddy 
■currents.  When  flat  strips  are  used,  the  laminations 
"SIlSTe  in  the  direction  of  the  leakage  flux  hnes.  It  is 
advisableTo  add  from  lo  to  15  per  cent  to  the  calculated 
PR  loss  when  the  currents  to  be  carried  are  large,  even 
after  reasonable  precautions  have  been  taken  to  avoid 
large  local  currents  by  subdividmg  the  conductors. 

The  mere  subdivision  of  a  conductor  of  large  cross- 
section  does  not  always  eliminate  the  injurious  effects 
of  local  currents  in  the  copper,  because,  unless  each  of  the 
several  conductors  that  are  joined  in  parallel  at  the  ter- 
minals does  not  enclose  the  same  amount  of  leakage 
Hux,  there  will  be  different  e.m.f.'s  developed  in  various 
sections  of  the  subdi\'ided  conductor,  and  consequent 
lack  of  uniformity  m  the  current  distribution.  This  ob- 
jection can  sometimes  be  overcome  by  giving  the  assem- 
bled conductor  (of  many  parallel  wires  or  strips)  a  half 
twTst,  and  so  changing  the  position  of  the  mdividual 
conductors  relatively  to  the  leakage  flux;  but,  in  any 
case,  once  this  cause  of  increased  copper  loss  is  recog- 
nized, it  is  generally  possible  to  dispose  and  join  together 
the  several  elements  of  a  compound  conductor  so  that 
the  leakage  flux  shall  affect  them  all  equally. 

21.  Efficiency.  The  output  of  a  single-phase  trans- 
former, in  watts,  is 

where  Es  is  the  secondary  terminal  voltage;  L,  the  sec- 
ondary current;  and  cos  d,  the  power  factor  of  the 
secondary  load.     The  percentage  efficiency  is  then: 

W^ 

^°°^t^_|-iron  losses + copper  losses' 


74  PRINCIPLES  OF  TRANSFORMER  DESIGN 

All-day  Efficiency.  The  all-day  efficiency  is  a  matter 
of  importance  in  coimection  with  distributing  trans- 
formers, because,  although  the  amount  of  the  copper 
loss  falls  off  rapidly  as  the  load  decreases,  the  iron  loss 
continues  usually  during  the  twenty-four  hours,  and  may 
be  excessive  in  relation  to  the  output  when  the  trans- 
former is  hghtly  loaded,  or  without  any  secondary  load, 
during  many  hours  in  the  day. 

What  is  understood  by  the  all-day  percentage  efficiency 
is  the  ratio  given  below,  the  various  items  being  cal- 
culated or  estimated  for  a  period  of  twenty-four  hours: 

looX  Secondary  output  in  watt-hours 

Sec.  watt-hrs.-f  watt-hrs.  iron  loss+watt-hrs.  copper  loss' 

It  is  m  order  that  this  quantity  may  be  reasonably  large 
that  the  iron  losses  in  distributing  transformers  are 
usually  less  than  in  power  transformers  designed  for  the 
same  maximum  output. 

Efficiency  of  Modern  Transformers.  The  alternating- 
current  transformer  is  a  very  efficient  piece  of  apparatus, 
as  shown  by  the  following  figures  which  are  an  indication 
of  what  may  be  expected  of  well-designed  transformers 
at  the  present  time. 

Full-load  Efficiencies  of  Small  Lighting  Trans- 
formers FOR  Use  on  Circuits  up  to  2200  Volts 

Output,  k.v'.a.  Efficiency  (per  cent) 

I From  94 . 1  to  96 

2 From  94 . 6  to  96 . 5 


5 From  95.5  to  97 

10 From  96 . 4  to  97 

20 From  97 . 2  to  98 

50 From  97.6  to  98 


EFFICIENCY  AND  HEATING  OF  TRANSFORMERS      75 

For  a  given  cost  of  materials,  the  efficiency  will  improve 
with  the  higher  frequencies,  and  a  transformer  designed 
for  a  frequency  of  25  would  rarely  have  an  efficiency 
higher  than  the  lower  limit  given  in  the  above  table, 
while  the  higher  figures  apply  mainly  to  transformers 
for  use  on  60-cycle  circuits. 

The  highest  efficiency  of  a  lighting  transformer  usually 
occurs   at   about   three-quarters  of  full  load.     Typical 
figures  for  a  5  k.v.a.  lighting  transformer  for  use  on  a 
50-cycle  circuit  are  given  below. 
Core  loss  =  46  watts. 
Copper  loss  (full  load)  =  114  watts. 
Calculated  efficiency  (100  per  cent  power  factor): 
At  full  load,  0.969. 
At  three-quarters  full  load,  0.9713. 
At  one-half  full  load,  0.9707. 
At  one-quarter  full  load,  0.9583. 

Full-load    Efficiencies    of    Power   Transformers 
FOR  Use  on  66,ooo-volt  Circuits 

(100  per  cent  power  factor) 

Output,  k.v.a.  Efficiency,  per  cent. 


400 From  97 

800 From  97 

1 200 From  97 

2000 From  98 

2600 From  08 


3  to  97 
7  to  98 
9  to  98 

1  to  98 

2  to  98 


The  manner  in  which  the  efficiency  of  large  power 
transformers  falls  off  with  increase  of  voltage  (involving 
loss  of  space  taken  up  by  insulation)  is  indicated  by  the 


76 


PRINCIPLES  OF  TRANSFORMER  DESIGN 


following  figures,  which  refer  to  looo  k.v.a.  single-phase 
units  designed  for  use  on  50-cycle  circuits. 


H.T.  Voltage. 


22,000. 
33,000. 
44,000. 
66,000. 
88,000. 
110,000. 


Full  Load  Efficiency 

(Appro.ximate) 

Per  cent. 


97 


The  figures  given  below  are  actual  test  data  showing 
the  performance  of  some  single-phase,  oil-insulated,  self- 
cooling,  power  transformers  recently  installed  in  a  hydro- 
electric generating  station  in  Canada: 

Output 400  k.v.a. 

Frequency /=6o 

Primary  volts 2,200 

Secondary  volts 22,000 

Core  loss 1,760  watts 

Full-load  copper  loss 3,55°  watts 

Exciting  current,  2.15  per  cent,  of  full-load  current. 
Temperature  rise  (by  thermometer)  after  contin- 
uous full-load  run,  36°  C. 
Efficiency  on  unity  power  factor  load : 

At  1.25  times  full  load. ...   98.57  per  cent 


At  full  load 98 

At  three-quarters  full  load .  98 

At  one-half  full  load 98 

At  one-quarter  full  load. .  .  98 


EFFICIENCY  AND  HEATING  OF  TRANSFORMERS      77 

It  should  be  stated  that  the  core  loss  in  these  trans- 
formers was  exceptionally  low,  being  only  0.44  per  cent  of 
the  k.v.a.  output.  The  core  losses  in  modern  trans- 
formers will  usually  he  between  the  limits  stated  below: 


K.v.a.  Output. 

Volts. 

Percentage  Core  Loss 

100  Xcore  loss,  watts, 
"rated  volt-ampere  output' 

Soo. . . . 
1000 

2000 . . .  .  < 
4000. . .  . 

22,000 
66,000 

22,000 
66,000 
110,000 

22,000 
66,000 
110,000 

66,000 
110,000 

0.7s  to  0.95 

1 . 0     to  I . 2 

0.6    too. 7 
0.7    to  I.O 
0.8    to  1. 15 

0.5    to  0.65 
0.55  to  0.7 
0.7    to  0.95 

0.5    too. 6 
0.6    to  0.7s 

The  core  losses  in  small  transformers  for  use  on  lighting 
circuits  up  to  2200  volts  are  usually  less  than  i  per  cent 
for  all  sizes  above  3  kw.  They  may  be  as  low  as  0.5 
per  cent  in  a  50  kw.  distributing  transformer,  and  as  high 
as  2.5  per  cent  in  a  i  kw.  transformer. 

The  frequency,  whether  25  or  60,  does  not  greatly 
influence  the  customary  allowance  for  core  loss. 

Efficiency  when  Power  Factor  oj  Load  is  Less  than  Unity. 
The  total  full-load  losses  (iron -j- copper)  may  be  ex- 
pressed as  a  percentage  of  the  k.v.a.  output.  Assume 
that  these  losses  are  equal  to  a(k.v.a.).  Then  at  any 
power  factor,  cos  6. 


78  PRINCIPLES  OF  TRANSFORMER  DESIGN 

T,£c  •  (k.v.a.)  cos  6 

Emciency  =  -r, ^^ r, ^, 

(k.v.a.)  cos  0+a(k.v.a.) 

cos  6 


cos  d+a 

Let  7]  stand  for  the  efficiency  at  unity  power  factor, 
then 

and 

l-r; 

whence  the  efficiency  at  any  power  factor,  cos  6,  is 
cos  6 


cos  e-\- 


(?)■ 


As  an  example,  calculate  the  full-load  efficiency  of  a 
transformer  on  a  load  of  0.75  power  factor,  given  that 
the  efficiency  on  unity  power  factor  is  0.969. 

The  ratio  of  the  total  losses  to  the  k.v.a.  output  is 

1—0.969 

a  = ^-^  =  0  02-2 

0.969 

whence  the  efficiency  at  0.75  power  factor  is 


0-7S 

— =  o-959. 

0.75+0.032 


EFFICIENCY  AND  HEATING  OF  TRANSFORMERS  ^JI^ 

22.  Temperature  of  Transformer  Windings.  Insu- 
lating materials  such  as  cotton  and  paper,  specially 
treated  with  insulating  compounds  or  immersed  in  oil, 
may  be  subjected  to  a  temperature  up  to,  but  not 
exceeding  105°  C.  The  hottest  spot  of  the  winding 
cannot  be  reached  by  a  thermometer,  and  it  is  therefore 
customary  to  add  15°  C.  to  the  temperature  registered 
by  a  thermometer  placed  at  the  hottest  accessible  part 
of  a  transformer  under  test.  The  room  temperature  is 
frequently  as  high  as  35°  C.  and  the  maximum  permis- 
sible rise  in  temperature  above  that  of  the  surroundmg 
air  may  be  arrived  at  as  follows: 

Permissible  hottest  spot  temperature .    105° 
Hottest  spot  correction 15 

Difference 90 

Assumed  room  temperature 35 

Difference  (  =  permissible  temperature 
rise) 55 

Thus,  under  the  worst  conditions  of  heating,  the  per- 
missible temperature  rise  should  not  exceed  55°  C.  when 
the  measurements  are  made  with  a  thermometer.  A 
more  rehable  means  of  arriving  at  transformer  tempera- 
tures is  to  calculate  these  from  resistance  measurements 
of  the  windings.  Such  measurements  usually  give  some- 
what higher  temperatures  than  when  thermometers  are 
used,  and  a  hottest  spot  correction  of  10°  C.  is  then  gen- 
erally  recognized   as   sufficient.     It   should   be  noted, 


80  PRINCIPLES  OF  TRANSFORMER  DESIGN 

however,  that  room  temperatures  of  40°  C.  are  not 
impossible,  and  it  is  therefore  customary  to  Hmit  the 
observed  rise  in  temperature  to  55°  C.  even  when  the 
resistance  method  of  measuring  temperatures  is  adopted. 

Transformers  are  usually  designed  to  withstand  an 
overload  of  two  hours'  duration  after  having  been  in 
continuous  operation  under  normal  full-load  conditions. 
Either  of  the  following  methods  of  rating  is  to  be  found 
in  modern  transformer  specifications: 

(i)  The  temperature  rise  not  to  exceed  40°  C.  on  con- 
tinuous operation  at  normal  load,  and  55°  C.  after  an 
additional  two  hours'  run  on  25  per  cent  overload. 

(2)  The  temperature  rise  not  to  exceed  35°  C.  on  con- 
tinuous operation  at  normal  load,  and  55°  C.  after  an 
additional  two  hours'  run  on  50  per  cent  overload. 

On  account  of  the  slow  heating  of  the  iron  core,  large 
oil-cooled  transformers  may  require  ten,  or  even  twelve 
hours  to  attain  the  final  temperature. 

23.  Heat  Conductivity  of  Insulating  Materials.  Be- 
fore discussing  the  means  by  which  the  heat  is  carried 
away  from  the  external  surface  of  the  coils,  it  will  be 
advisable  to  consider  how  the  designer  may  predetermine 
approximately  the  difference  in  temperature  between 
the  hottest  spot  and  the  external  surface  of  the  windings. 
Calculations  of  internal  temperatures  cannot  be  made 
very  accurately;  but  the  nature  of  the  problem  is  indi- 
cated by  the  following  considerations: 

Fig.  28  is  supposed  to  represent  a  section  through  a 
very  large  flat  plate,  of  thickness  /,  consisting  of  any 
homogeneous  material.  Assume  a  difference  of  tem- 
perature of  Td  =  {T-To)°C.  to  be  maintained  between 


EFFICIENCY  AND  HEATING  OF  TRANSFORMERS      81 


the  two  sides  of  the  plate,  and  calculate  the  heat  flow 
(expressed  in  watts)  through  a  portion  of  the  plate  of 
area  wXl.  The  resistance  offered  by  the  material  of 
the  plate  to  the  passage  of  heat  may  be  expressed  in 
thermal  ohms,  the  thermal  ohm  being  defined  as  the 
thermal  resistance  which  causes  a  drop  of  i°  C.  per  watt 


1 

1- 

1 

I 

1 

i 

Watts =W 

3 

Fig.  28. — Diagram  Illustrating  Heat  Flow  through  Flat  Plate. 

of  heat  flow;  or,  if  Rn  is  the  thermal  resistance  of  the  heat 
path  under  consideration, 

T 

(19) 


7?  —^'^ 


which  permits  of  heat  conduction  problems  being  solved 
by  methods  of  calculation  similar  to  those  used  in  con- 
nection with  the  electric  circuit. 


82 


PRINCIPLES  OF  TRANSFORMER  DESIGN 


Let  k  be  the  heat  conductivity  of  the  material,  ex- 
pressed in  watts  per  inch  cube  per  degree  Centigrade 
difference  of  temperature  between  opposite  sides  of  the 


\ 

U-a^ 

fdx 

___. 

! 

^ t 

2 

— > 

Y 
Fig.  29. — Heat  Conductivity:  Heat  Generated  Inside  Plate. 


cube,  then  the  watts  of  heat  flow, crossing  the  area 
{wXl)  square  inches,  as  indicated  in  Fig.  28  is 


W- 


m^- 


(20) 


Fig.  29  illustrates  a  similar  case,  but  the  heat  is  now 
supposed  to  be  generated  in  the  mass  of  the  material 
itself.     We  shall  still  consider  the  plate  to  be  very  large 


EFFICIENCY  AND  HEATING  OF  TRANSFORMERS      83 

relatively  to  the  thickness,  so  that  the  heat  flow  from 
the  center  outward  will  be  in  the  direction  of  the  hori- 
zontal dotted  lines,  A  uniformly  distributed  electric 
current  of  density  A  amperes  per  square  inch  is  supposed 
to  be  flowing  to  or  from  the  observer,  and  the  highest 
temperature  will  be  on  the  plane  YY'  passing  through 
the  center  of  the  plate.  Assuming  this  plate  to  be  of 
copper  with  a  resistivity  of  0.84X10-6  ohms  per  inch 
cube  at  a  temperature  of  about  80°  C,  the  watts  lost  in 
a  section  of  area  (xXw)  sq.  in.  and  length  I  in.  will  be 

PF:,  =  (Axw)2  X  0.84  X 10-6  X— 

xw 

=  o.84Xio-'^A'^wlx (21) 

By  adapting  Formula  (20)  to  this  particular  case,  the 
difference  of  temperature  between  the  two  sides  of  a 
section  dx  in.  thick  is  seen  to  be 


dTa  =  W.X    ^"^ 


whence, 


wlXk 
0.84A2/2 


Xio^;^ 


degrees  Centigrade.  .     .     (22) 


The  value  of  k  for  copper  is  about  10  watts  per  inch 
cube  per  degree  Centigrade. 

The  problem  of  applying  these  principles  to  the  prac- 
tical case  of  a  transformer  coil  is  complicated  by  the  fact 


84 


PRINCIPLES  OF  TRANSFORMER  DESIGN 


that  the  heat  does  not  travel  along  parallel  paths  as  in 
the  preceding  examples,  and,  further,  that  the  thermal 
conductivity  of  the  built-up  coil  depends  upon  the  rel- 
ative thickness  of  copper  and  insukting  materials,  a 
relation  which  is  usually  different  across  the  layers  of 
windmg  from  what  it  is  in  a  direction  parallel  to  the 
layers. 

A 


Fig.  30. — Diagram  Illustrating  Heat  Paths  in  a  Transformer  Coil  of 
Rectangular  Cross-section. 


Fig.  30  represents  a  section  through  a  transformer 
coil  wound  with  layers  of  wire  in  the  direction  OA; 
the  number  of  layers  being  such  as  to  produce  a  total 
depth  of  winding  equal  to  twice  OB.  The  whole  of  the 
outside  surface  of  this  coil  is  supposed  to  be  maintained 
at  a  constant  temperature  by  the  surrounding  oil  or  air. 
In  other  words,  it  is  assumed  that  there  is  a  constant 
difference  of  temperature  of  Td  degrees  between  the 
hottest  spot  (supposed  to  be  at  the  center  0)  and  any 
point  on  the  surface  of  the  coil. 


EFFICIENCY  AND  HEATING  OF  TRANSFORMERS      85 

The  heat  generated  in  the  mass  of  material  is  thought 
of  as  traveling  outward  through  the  walls  of  successive 
imaginary  spaces  of  rectangular  section  and  length  I 
(measured  perpendicularly  to  the  plane  of  the  section 
shown  in  Fig.  30),  as  indicated  in  the  figure,  where 
CDEF  is  the  boundary  of  one  of  these  imaginary  spaces, 
the  walls  of  which  have  a  thickness  dx  in  the  direction 

OA,  and  a  thickness  dx  I——]  in  the  direction  OB. 


© 


According  to  Formula  (19),  we  may  say  that  the  dif- 
ference of  temperature  between  the  inner  and  outer 
boundaries  of  this  imaginary  wall  is  dTa  =  hea,t  loss,  in 
watts,  occurring  in  the  space  CDEFX  the  thermal 
resistance  of  the  boundary  walls. 

It  is  proposed  to  consider  the  heat  flow  through  the 
portion  of  the  boundary  surface  of  which  the  area  is 
CDEF XL 

If  Wx  stands  for  the  watts  passing  through  this  area, 
we  may  write 

dTa  =  WxX 


2DElka         2CDlh 


dx         J 
dx 


which  simplifies  into 


dTa  =  W.      .     .^^.2 y  ;    .     .      (23) 


86  PRINCIPLES  OF  TRANSFORMER  DESIGN 

In  order  to  calculate  Wx  it  is  necessary  to  know  not 
only  the  current  density,  A,  but  also  the  space  factor,  or 
ratio  of  copper  cross-section  to  total  cross- section. 

Let  a  stand  for  the  thickness  of  copper  per  inch  of  total 
thickness  of  coil  measured  in  the  direction  OA ;  and  let 
b  stand  for  a  similar  quantity  measured  in  the  direction 
OB ;  the  space  factor  is  then  (aXb),  and 

W^:,=  |AX2:rX2a;f^)xa^'J  0.84X10-6 . 

2xX2x{^^)ab 


Inserting  this  value  of  Wx  in  (23),  and  making  the 
necessary  simplij&cations,  we  get 

,^  o.S4A^ab 


A^Mmf] 


whence,  by  integration  between  the   limits  x  =  o  and 
x  =  OA, 

Ta  = r /HA. 21  deg.  Cent.     .     (24) 


{...Ql 


Except  for  the  obvious  correction  due  to  the  intro- 
duction of  the  space  factor  (ab),  the  only  difference 
between  this  formula  and  Formula  (22)  is  that  the 
thermal  conductivity,  instead  of  being  ka,  as  it  would 
be  if  the  heat  flow  were  m  the  direction  OA  only,  is 


EFFICIENCY  AND  HEATING  OF  TRANSFORMERS      87 

replaced  by  the  quantity  in  brackets  in  the  denominator 
of  Formula  (24).  This  quantity  may  be  thought  of  as  a 
fictitious  thermal  conductivity  in  the  direction  OA, 
which,  being  greater  than  ka,  provides  the  necessary  cor- 
rection due  to  the  fact  that  heat  is  being  conducted  away 
in  the  direction  OB,  thus  reducing  the  difference  of  tem- 
perature between  the  points  O  and  A. 

Calculation  of  ka  and  h. 

Let  kc  and  kt,  respectively,  stand  for  the  thermal  con- 
ductivity of  copper  and  insulating  materials  as  used  in 
transformer  construction.  The  numerical  values  of 
these  quantities,  expressed  in  watts  per  inch  cube  per 
degree  Centigrade,  are  kc  =  10  and  kt  =  o.cx333.  ^t  follows 
that 


"    a      (i—a)     a+300o(i— o)'       •     '     ^  5) 


and  similarly. 


^^  =  ^+3000(1-6)'      •     •    •    •     (26) 


where  a  and  h  are  the  thickness  of  copper  per  inch  of 
coil  in  the  directions  OA  and  OB,  respectively,  as  pre- 
viously defined. 

Example.  Suppose  a  transformer  coil  to  be  wound 
with  0.25X0.25  in.  square  copper  wire  insulated  with 
cotton  o.oi  in.  thick,  and  provided  with  extra  insulation 
of  0.008  in.  fullerboard  between  layers.  There  are 
twelve  layers  of  wire  and  seven  wires  per  layer.  Assume 
the  current  density  to  be  1400  amperes  per  square  inch, 


88  PRINCIPLES  OF  TRANSFORMER  DESIGN 

and  calculate  the  hottest  spot  temperature  if  the  outside 
surface  of  the  coil  is  maintained  at  75°  C. 

0.25  0-25 

a  =  - — =0.926;    b  = 0=0-9;   whence  space  factor 

0.27        ^     '         0.278       ^'  ^ 

(a6)  =0.833. 

By  Formulas  (25)  and  (26),  ^0  =  0.0448,  and  ^6  =  0.0332; 
OA  =3.5X0.27=0.945,  and  05  =  6X0.278  =  1.67  in. 

By  Formula  (24), 

^  0.84(1400)^X0.833(0.945)2 

1  d  = r 1 wn  =11    Cent., 

2  XIO^' [0.0448 +0.033 2  (^^j   J 

and  the  hottest  spot  temperature  =  75  +  11  =86°  C. 

24.  Cooling  Transformers  by  Air  Blast.  Before  the 
advantages  of  oil  insulation  had  been  realized,  trans- 
formers were  frequently  enclosed  in  watertight  cases, 
the  metal  of  these  cases  being  separated  from  the  hot 
parts  of  the  transformer  by  a  layer  of  still  air.  This 
resulted  either  in  high  temperatures  or  in  small  kilowatts 
output  per  pound  of  material.  Air  insulation  is  still 
used  in  some  designs  of  large  transformers  for  pressures 
up  to  about  33,000  volts;  but  efficient  cooling  is  ob- 
tained by  forcing  the  air  around  the  windings  and 
through  ducts  provided  not  only  between  the  coils, 
but  also  between  the  coils  and  core,  and  between  sec- 
tions of  the  core  itself. 

Since  all  the  heat  losses  which  are  not  radiated  from 
the  surface  of  the  transformer  case  must  be  carried  away 
by  the  air  blast,  it  is  a  simple  matter  to  calculate  the 


EFFICIENCY  AND  HEATING  OF  TRANSFORMERS      89 

weight  (or  volume)  of  air  required  to  carry  away  these 
losses  with  a  given  average  increase  in  temperature  of 
outgoing  over  ingoing  air.    ' 

A  cubic  foot  of  air  per  minute,  at  ordinary  atmospheric 
pressures,  will  carry  away  heat  at  the  rate  of  about  0.6 
watt  for  every  degree  Centigrade  increase  of  tempera- 
ture. Thus,  if  the  difference  of  temperature  between 
outgoing  and  ingoing  air  is  10°  C,  the  quantity  of  air 
which  must  pass  through  the  transformer  for  every  kilo- 
watt of  total  loss  that  is  not  radiated  from  the  surface 
of  the  case,  is 


0=— 7 =  166  cu.  ft.  per  minute. 

0.6X10 


If  the  average  increase  in  temperature  of  the  air  is 
from  10  to  15°  C,  the  actual  surface  temperature  rise  of 
the  windmgs  may  be  from  40  to  50°  C;  the  exact  figure 
being  difficult  to  calculate  since  it  will  depend  upon  the 
size  and  arrangement  of  the  air  ducts.  The  temperature 
of  the  coils  is  influenced  not  only  by  the  velocity  of  the 
air  over  the  heated  surfaces,  but  also  by  the  amount  of 
the  total  air  supply  which  comes  into  intimate  contact 
with  these  surfaces.  With  air  passages  about  |  in. 
wide,  and  an  average  air  velocity  through  the  ducts 
rangmg  from  300  to  600  ft.  per  minute,  the  temperature 
rise  of  the  coil  surfaces  will  usually  be  from  four  to 
eight  times  the  rise  in  temperature  of  the  circulating 
air.  Thus,  although  it  is  not  possible  to  predetermine 
the  exact  quantity  of  air  necessary  to  maintain  the 


90  PRINCIPLES  OF  TRANSFORMER  DESIGN 

transformer  windings  at  a  safe  temperature,  this  may  be 
expressed  approximately  as: 


Cubic  feet  of  air  per  minute 
for  50°  C.  temperature  rise 
of  coil  surface 


0.6  X%^' 

=  0.2{W,-Wr),      .       (27) 


where  T^<  =  total  watts  lost  in  transformer;  and 

PFr  =  portion  of  total  loss  dissipated  from  surface 
of  tank. 

The  latter  quantity  may  be  estimated  by  assuming 
the  temperature  of  the  case  to  be  about  10°  C.  higher 
than  that  of  the  surrounding  air,  and  calculating  the 
watts  radiated  from  the  case  with  the  aid  of  the  data  in 
the  succeeding  article. 

Assuming  Wr  to  be  25  per  cent  of  Wt,  the  Formula  (27) 
indicates  that  about  150  cu.  ft.  of  air  per  minute  per 
kilowatt  of  total  losses  would  be  necessary  to  limit  the 
temperature  rise  of  the  coils  to  50°  C.  With  poorly 
designed  transformers,  and  also  in  the  case  of  small 
units,  the  amount  of  air  required  may  be  appreciably 
greater. 

It  is  true  that,  in  turbo-generators,  an  allowance  of 
100  cu.  ft.  per  minute  per  kilowatt  of  total  losses,  is  gen- 
erally sufficient  to  limit  the  temperature  rise  to  about 
50°  C. ;  but,  owing  to  the  churning  of  the  air  due  to  the 
rotation  of  the  rotor,  it  would  seem  that  the  necessary 
supply  of  air  is  smaller  for  turbo-generators  than  for 
transformers. 


EFFICIENCY  AND  HEATING  OF  TRANSFORMERS      91 

Filtered  air  is  necessary  in  connection  with  air-blast 
cooling;  otherwise  the  ventilating  ducts  are  liable  to 
become  choked  up  with  dirt,  and  high  temperatures  will 
result.  Wet  air  filters  are  very  satisfactory  and  desir- 
able, provided  the  amount  of  moisture  in  the  air  passing 
through  the  transformers  is  not  sufficient  to  cause  a 
deposit  of  water  particles  on  the  coils.  Air  containing 
from  I  to  3  per  cent  of  free  water  in  suspension  is  a  much 
more  effective  coohng  medium  than  dry  air.  It  would 
probably  be  inad\nsable  to  use  anything  but  dry  air  in 
contact  with  extra-high  voltage  apparatus;  but  trans- 
formers for  very  high  pressures  are  not  designed  for 
air-blast  cooling.* 

25.  Oil-immersed  Transformers — Self  Cooling.  The 
natural  circulation  of  the  oil  as  it  rises  from  the  heated 
surfaces  of  the  core  and  windings,  and  flows  downward 
near  the  sides  of  the  containing  tank,  will  lead  to  a  tem- 
perature distribution  generally  as  indicated  in  Fig.  31. 
The  temperature  of  the  oil  at  the  hottest  part  (close  to 
the  windings  at  the  top  of  the  transformer)  will  be  some- 
what higher  than  the  maximum  temperature  of  the  tank, 
which,  however,  will  be  hotter  in  the  neighborhood  of  the 
oil  level  than  at  other  parts  of  its  surface.  The  average 
temperature  of  the  cooling  surface  in  contact  with  the 
air  bears  some  relation  lo  the  highest  oil  temperature, 
and,  since  this  relation  does  not  vary  greatly  with 
different  designs  of  transformer,  or  case,  a  curve  such 

*  Some  useful  data  on  the  relative  cooling  effects  of  moist  and  dry  air, 
together  with  test  figures  relating  to  a  12-kw.  air-cooled  transformer, 
will  be  found  in  Mr.  F.  J.  Teago's  paper  "  Experiments  on  Air-blast 
Cooling  of  Transformers,"  in  the  Jour.  Inst.  E.  E.,  May  i,  1914  ,Vol.  52, 
page  563. 


92 


PRINCIPLES  OF  TRANSFORMER  DESIGN 


as  Fig.  32  may  be  used  for  calculating  the  approximate 
tank  area  necessary  to  prevent  excessive  oil  temperatures. 
The  oil  temperature  rise  referred  to  in  Fig.  32  is  the 
difference  in  degrees  Cent,  between  the  temperature 
of  the  hottest  part  of  the  oil  and  the  air  outside  the  tank. 


Temperature  of  covei 


v/afl'l'<'Ul'^l'A^l'^^^^^l'l'^/^/^^^//A'tufuu/tZ 


Fig.  31.— Distribution  of  Temperature  with  Transformer  Immersed  in 
Oil. 


This  will  be  somewhat  greater  than  the  temperature  rise 
of  any  portion  of  the  transformer  case;  but  the  curve 
indicates  the  (approximate)  number  of  watts  that  can 
be  dissipated — by  radiation  and  air  currents — per  square 
inch  of  tank  surface.  The  curve  is  based  on  average 
figures  obtained  from  tests  on  tanks  with  smooth  surjaces 


EFFICIENCY  AND  HEATING  OF  TRANSFORMERS      93 


i 


"0  0.05  0.1  0.15         0.2  0.25  0.3  0.35         0.4 

^="Watt8  dissipated  peraq.  in.  of  tank  surface 

Fig.  32.— Curve  for  Calculating  Cooling  Area  of  Transformer  Tanks. 


94  PRINCIPLES  OF  TRANSFORMER  DESIGN 

(not  corrugated),  the  surface  considered  being  the  total 
area  of  the  (vertical)  sides  plus  one-half  the  area  of  the 
lid.  The  coohng  effect  of  the  bottom  of  the  tank  is 
practically  negUgible,  and  is  not  to  be  included  in  the 
calculations. 

Example.  What  will  be  the  probable  maximum  tem- 
perature rise  of  the  oil  in  a  self-cooling  transformer  with 
a  total  loss  of  1200  watts,  the  tank — of  sheet-iron  with- 
out corrugations — measuring  2  ft.X2  ft.  X 3.5  ft.  high? 

The  surface  for  use  in  the  calculations  is  5  =  (3.5X8) 
-1-2=30  sq.  ft,  whence 


1200  „ 

=  0.278, 


30  X 144 

which,  according  to  Fig.  32,  indicates  a  43°  C.  rise  of 
temperature  for  the  oil. 

The  temperature  of  the  windings  at  the  hottest  part  of 
the  surface  in  contact  with  the  oil  might  be  from  5  to  10° 
C.  higher  than  the  maximum  oil  temperature  as  meas- 
ured by  thermometer.  Assume  this  to  be  7°  C.  Assume 
also  that  the  room  temperature  is  35°  C,  and  that  the 
difference  of  temperature  {Ta)  between  the  coil  surface 
and  the  hottest  spot  of  the  windings — as  calculated  by 
the  method  explained  in  Art.  23 — is  13°  C.  Then  the 
hottest  spot  temperature  in  the  transformer  under  con- 
sideration would  be  about  3 5 +43  4- 7 -f  13  =  98°  C. 

26.  Effect  of  Corrugations  in  Vertical  Sides  of  Con- 
taining Tank.  The  cooling  surface  in  contact  with  the 
air  may  be  increased  by  using  corrugated  sheet-iron 
tanks  in  place  of  tanks  with  smooth  sides.     It  must  not, 


EFFICIENCY  AND  HEATING  OF  TRANSFORMERS      95 

however,  be  supposed  that  the  temperature  reduction 
will  be  proportional  to  the  increase  of  tank  surface 
provided  in  this  manner;  the  watts  radiated  per  square 
inch  of  surface  of  a  tank  with  corrugated  sides  will  always 
be  appreciably  less  than  when  the  tank  has  smooth  sides. 
Not  only  is  the  surface  near  the  bottom  of  the  corruga- 
tions less  effective  in  radiating  heat  than  the  outside 
portions;  but  the  depth  and  pitch  of  the  corrugations 
will  affect  the  (downward)  rate  of  flow  of  the  oil  on  the 
inside  of  the  tank,  and  the  (upward)  convection  cur- 
rents of  air  on  the  outside. 

It  is  practically  impossible  to  develop  formulas  which 
will  take  accurate  account  of  all  the  factors  involved,  and 
recourse  must  therefore  be  had  to  empirical  formulas 
based  on  available  test  data  together  with  such  reason- 
able assumptions  as  may  be  necessary  to  render  them 
suitable  for  general  application. 

If  X  is  the  pitch  of  the  corrugations,  measured  on  the 
outside  of  the  tank,  and  I  is  the  surface  width  of  material 
per  pitch  (see  the  sketch  in  Fig.  T)^),  the  ratio  of  the 
actual  tank  surface  to  the  surface  of  a  tank  without 

corrugations  is  -.    The  heat  dissipation  will  not  be 

A 

in  this  proportion  because,  although  the  cooling  effect 
will  increase  as  /  is  made  larger  relatively  to  X,  the 
additional  surface  becomes  less  and  less  effective  in 
radiating  heat  as  the  depth  of  the  corrugations  increases 
without  a  corresponding  increase  in  the  pitch.  It  is 
convenient  to  think  of  the  surface  of  an  equivalent 
smooth  tank  which  will  give  the  same  temperature 
rise  of  the  oil  as  will  be  obtained  with  the  actual  tank. 


96 


PRINCIPLES  OF  TRANSFORMER  DESIGN 


If    we    apply    a    correction    to    the    actual    pitch,    X, 

and   obtain   an   equivalent   pitch,    X^,    the   ratio   k  =  ~ 

X 

is  a  factor  by  which  the  tank  surface  (neglecting  corru- 


S 


1.8 


1.2 


\  < ^ > 


0.3 


0.4 


O.G 


0.7 


Fig.  ^2>- — Curve  Giving  Factor  k  for  Calculating  Equivalent  Cooling 
Surface  of  Tanks  with  Corrugated  Sides. 


gations)  must  be  multiplied  in  order  to  obtain  the 
equivalent  or  efeclive  surface.  If  all  portions  of  the 
added  surface  were  equally  effective  in  radiating  heat, 
no  correcting  factor  would  be  required,  and  the  equiva- 
lent pitch  would  be  obtained  by  adding  to  X  the  quan- 


EFFICIENCY  AND  HEATING  OF  TRANSFORMERS      97 

tity  (/  — X);  but  since  a  modifying  factor  is  needed, 
the  writer  proposes  the  formula 

^-^+('-^)(4^).  •  •  •  ■  (^«) 

wherein  the  additional  surface  provided  by  the  cor- 
rugations is  reduced  in  the  ratio  - —   which  becomes 

unity  when  /  =  X.  A  modifying  factor  of  this  form  not 
only  seems  reasonable  on  theoretical  grounds,  but  it 
is  required  in  an  empirical  formula  based  on  available 
experimental  data.     It  follows  that 


,     X,         ,  2(/-X)  ,     V 


.,  X 
or,  if-  =  «, 


ife=I  +  2 


(S) ^30) 


Values  of  k,  as  obtained  from  this  formula  for  different 
values  of  n,  may  be  read  off  the  curve  Fig.  ;^7,. 

Example.  What  would  have  been  the  temperature 
rise  of  the  oil  if,  instead  of  the  smooth-side  tank  of  the 
preceding  example  (Art.  25),  a  tank  of  the  same  external 
dimensions  had  been  provided  with  corrugations  2  in. 
deep,  spaced  i|  in.  apart? 

The    approximate    value    of   /    is    1.25+4  =  5.25    in. 

12^ 
Whence  w  =  ^-^=  0.238;    and  from  the  curve.  Fig.  30, 

^  =  2.23. 


98  PRINCIPLES  OF  TRANSFORMER  DESIGN 

The  equivalent  tank  surface  is  5=  (3.5X8X2.23) +  2 
=  64.5  sq.  ft.,  whence 


1200 

20  = =  0.129, 

64.5X144 


which,  according  to  Fig.  32,  indicates  a  27°  C.  rise  of 
temperature,  as  compared  with  43°  C.  with  the  smooth- 
surface  tank  of  the  same  outside  dimensions. 

27.  Effect  of  Overloads  on  Transformer  Tempera- 
tures. Since  the  curve  of  Fig.  32  is  not  a  straight  line, 
it  follows  that  the  watts  dissipated  per  square  inch  of 
tank  surface  are  not  directly  proportional  to  the  differ- 
ence between  the  oil,  and  room,  temperatures.  The 
approximate  relation,  according  to  this  curve,  is 

Temperature  rise  =  constant  Xtc;"^,     .     .     (31) 

which  may  be  used  for  calculating  the  temperature  rise 
of  a  self-cooling  oil-immersed  transformer  when  the  tem- 
perature rise  under  given  conditions  of  loading  is  known. 
Example.  Given  the  following  particulars  relating  to 
a  transformer: 

Core  loss  =  100  watts. 

Copper  loss  (full  load)  =  200, 

Final  temp,  rise  (full  load)  of  the  oil  =  35°  C. 

Calculate  the  final  temperature  rise  after  a  continuous 
run  at  20  per  cent  overload. 

For  an  increase  of  20  per  cent  in  the  load,  the  copper 


EFFICIENCY  AND  HEATING  OF  TRANSFORMERS      99 

loss  is  2ooX  (1.2)2  =  288  watts;    whence,  according  to 
Formula  (31): 

rr              ,          '              ^/288  +  iooY-«       on 
Temperature  rise  =  ^  t;  X   =41    C.  approx. 

\200+I00/ 


The  calculation  of  temperature  rise  resulting  from  an 
overload  of  short  duration  is  not  so  simple.  It  is  neces- 
sary to  take  account  of  the  specific  heat  of  the  materials, 
especially  the  oil,  because  the  heat  units  absorbed  by 
the  materials  have  not  to  be  radiated  from  the  tank 
surface,  and  the  calculated  temperature  rise  would  be 
too  high  if  this  item  were  neglected. 

The  specific  heat  of  a  substance  is  the  number  of  calor- 
ies required  to  raise  the  temperature  of  i  gram  1°  C, 
the  specific  heat  of  water  being  taken  as  unity. 

The  specific  heat  of  copper  is  0.093,  ^^'^  ^^^  ^^  average 
quahty  of  transformer  oil,  it  is  0.32. 

One  gram-calorie  {i.e.,  the  heat  necessary  to  raise  the 
temperature  of  i  gram  of  water  1°  C.)  =4.183  joules  (or 
watt-seconds).  Also,  i  lb.  =453.6  grams.  It  follows 
that  the  amount  of  energy  in  watt-seconds  necessary  to 
raise  Mc  pounds  of  copper  T°  C.  is 

WattsXtime  in  seconds  =  4. 183X0.093X453.6  McT 
=  177  McT  (for  copper). 

Similarly,  if  we  put  Mo  for  the  weight  of  oil,  in  pounds, 
and  replace  the  figure  0.093  by  0.32,  we  get 

WattsXtime  in  seconds  =  610  MoT  (for  oil). 


100  PRINCIPLES  OF  TRANSFORMER  DESIGN 

In  the  case  of  an  overload  after  the  transformer  has 
been  operating  a  considerable  length  of  time  on  normal 
full  load,  all  the  additional  losses  occur  in  the  copper 
coils,  and  it  is  generally  permissible  to  neglect  the 
heat  absorption  by  the  iron  core.  We  shall,  therefore, 
assume  that  the  additional  heat  units  which  are  not 
absorbed  by  the  copper  pass  into  the  oil,  and  that  the 
balance,  which  is  not  needed  to  heat  up  the  oil,  must  be 
dissipated  by  radiation  and  convection  from  the  sides 
of  the  containing  tank.  It  will  greatly  simplify  the  cal- 
culations if  we  further  assume  that  the  watts  dissipated 
per  square  inch  of  tank  surface  per  degree  difference  of 
temperature  are  constant  over  the  range  of  temperature 
involved  in  the  problem.  (By  estimating,  the  average 
temperature  rise,  and  finding  w  on  the  curve.  Fig.  32, 

w 
a  suitable  value  for  the  quantity  —  may  be  selected.) 

If  Pr«  =  total  watts  lost  (iron + copper),  the  total 
energy  loss  in  the  interval  of  time  dt  second 
is  Widt. 

If  the  increase  of  temperature  during  this  interval  of 
time  is  dx  degree  Centigrade,  the  heat  units  absorbed 
by  the  copper  coils  and  the  oil  are  Kdx,  where  Ks  = 
(177MC+610M0).* 

The  difference  between  these  two  quantities  represents 
the  number  of  joules,  or  watt-seconds,  of  energy  to  be 

*  In  order  to  simplify  the  calculations,  it  has  been  (incorrectly)  assumed 
that  the  temperature  rise  of  the  copper  is  the  same  as  that  of  the  oil. 
It  will,  of  course,  be  somewhat  greater;  but  since  the  heat  absorbed  by 
the  copper  is  small  compared  with  that  absorbed  by  the  oil,  this  assumpn 
tion  will  not  lead  to  an  error  of  appreciable  magnitude. 


EFFICIENCY  AND  HEATING  OF  TRANSFORMERS    101 

radiated  from  the  tank  surface  during  the  interval  of 
time  dt  second;  whence, 

Wtdt-Ksdx  =  Krxdt,     ....     (32) 

where  i^r  =  tank  surface  in  square  inches  X  radiation 
coefficient,  in  watts  per  square  inch  per  1°  C.  rise,  and 
a;  =  the  initial  oil  temperature  rise  (which  has  been 
increased  by  the  amount  dx). 

Equation  (32)  may  be  put  in  the  form 

di  Ks 


dx      Wt-KrX 

The  limits  for  x  are  the  initial  oil  temperature  Tq 
and  the  final  oil  temperature  Tt,  which  is  reached  at 
the  end  of  the  time  /.     Therefore, 


^   dx 

KtX 


If  time  is  expressed  in  minutes,  and  common  logs, 
are  used,  we  have, 


102  PRINCIPLES  OF  TRANSFORMER  DESIGN 

In  order   to  facilitate  the  use  of  this  formula,   the 
meaning  of  the  symbols  is  repeated  below: 

Wt  =  total  watts  lost  (iron + copper); 

i!Cr  =  5  X  radiation    coefficient    expressed    in    watts    per 

square  inch  per  i°  C.  rise  of  temperature  of  the 

oil; 
where  5  =  tank  surface   in    square    inches,    as   de- 
fined  in   Art.    25,    corrected    if    necessary    for 

corrugations  (Art.  26). 
Ks  =  i77Mc+6ioMq;  • 

where  Mc  =  weight  of  copper  (pounds) ; 

and     Mo  =  weight  of  oil  (pounds) ; 
7*0  =  initial  temperature  of  oil  (degrees  C); 
r<  =  temperature  of  oil  (degrees  C.)  after  the  overload 

(producing  the  total  losses  Wt)  has  been  on  for  tm 

minutes. 

Example.     Using  the  data  of  the  preceding  example, 
the  full-load  conditions  are: 

Core  loss  =100  watts; 

Copper  loss  =200  watts; 

Temperature  rise  =  35°  C. 

Referring  to  Fig.  29,  the  value  for  w  for  a  temper- 
ature rise  of  35°  is  0.193,  from  which  it  follows  that 

«.      .  1         r        '     r^     100+200 

the  effective  tank  surface  is  S  = =  i5So  sq.  m. 

0.193 

Given  the  additional  data: 

Weight  of  copper  =  65  lb., 
Weight  of  oil        =  140  lb. 


EFFICIENCY  AND  HEATING  OF  TRANSFORMERS    103 

calculate  the  time  required  to  raise  the  oil  from  To  =  35°  C. 
to  Tt  =  4S°  C.  on  an  overload  of  50  per  cent. 
The  copper  loss  is  now  200X  (1.5)^  =  450  watts,  whence 

Wt  =  100+450  =  550  watts. 

The  cooHng  coefficient  (from  curve,  Fig.  32),  for  an 

r   35+45          o    r-      '     °-242 
average  temperature  rise  of  =  40     C,  is  

=0.00606,  whence, 

Xr  =  1 5  50  X  0.00606  =  9.4 ; 

Zs  =  (i77X65)  +  (6ioXi4o)  =97,000; 
and,  by  Formula  (34), 

97,000     ,  9.4        ^^    \  .         ^ 

'"  =  ;65<^'°S    S^    =95.5mmutes. 

28.  Self-cooling    Transformers    for    Large    Outputs. 

The  best  way  to  cool  large  transformers  is  to  provide 
them  with  pipe  coils  through  which  cold  water  is  circu- 
lated, or,  alternatively,  to  force  the  oil  through  the  ducts 
and  provide  means  for  cooling  the  circulating  oil  outside 
the  transformer  case.  When  such  methods  cannot  be 
adopted — as  in  most  outdoor  installations  and  other 
sub-stations  without  the  necessary  machinery  and  at- 
tendants— the  heat  from  self-cooling  transformers  of 
large  size  is  dissipated  by  providing  additional  cooling 
surface  in  the  form  of  tubes,  or  flat  tanks  of  small  volume 


104 


PRINCIPLES  OF  TRANSFORMER  DESIGN 


and  large  external  surface,  connected  to  the  outside  of  a 
central  containing  tank.  Unless  test  data  are  available 
in  connection  with  the  particular  design  adopted,  judg- 
ment is  needed  to  determine  the  effective  cooling  surface 
(see  Art.  26)  in  order  that  the  curve  of  Fig.  32 — or  such 
cooling  data  as  may  be  available  for  smooth-surface 
tanks — may  be  used  for  calculating  the  probable  tem- 
perature rise. 

In  the  tubular  type  of  transformer  tank  which  is  pro- 
vided with  external  vertical  tubes  cormecting  the  bottom 
of  the  tank  to  the  level, 
near  the  oil  surface,  where 
the  temperature  is  highest 
(as  roughly  illustrated  by 
Fig.  34),  the  tubes  should 
be  of  fairly  large  diameter 
with  sufficient  distance  be- 
tween them  to  allow  free 
circulation  of  the  air  and 
efficient  radiation.  It  is 
not  economical  to  use  a 
very  large  number  of  small 
tubes  closely  spaced  with 
a  view  to  obtaining  a  large 
cooling  surface,  because  the 

Fig.    34.— Transformer    Case    with       ,  .  l.    •       1  i. 

Tubes  to  Provide  Additional  Cool- e^tra  surface  ob tamed  by 

ing  Surface.  such     means     is     not     as 

effective    as    when    wider 

spacing  is  used.     If  the  added  pipe  surface,  Ap,  is  1.5 

times  the  tank  surface,  At,  without  the  pipes,  the  effective 

cooling  surface  will  be  about  S  =  (Ai-hAp)Xo.g;  but, 


EFFICIENCY  AND  HEATING  OF  TRANSFORJMERS    105 

with  a  greatly  increased  surface  obtained  by  reducing 
the  spacing  between  the  pipes,  the  correction  factor 
might  be  very  much  smaller  than  0.9. 

29.  Water-cooled  Transformers.  The  cooling  coil 
should  be  constructed  preferably  of  seamless  copper  tube 
about  ij  in.  diameter,  placed  near  the  top  of  the  tank, 
but  below  the  surface  of  the  oil.  If  water  is  passed 
through  the  coil,  heat  will  be  carried  away  at  the  rate  of 
1000  watts  for  every  3I  gals,  flowing  per  minute  when  the 
difference  of  temperature  between  the  outgoing  and 
ingoing  water  is  1°  C.  Allowing  0.25  gal.  per  minute, 
per  kilowatt,  the  average  temperature  rise  of  the  water 

will  be  ^^^  =15°  C.     The  temperature  rise  of  the  oil  is 
0.25 

considerably  greater  than  this:  it  will  depend  upon  the 
area  of  the  coil  in  contact  with  the  oil  and  the  condition 
of  the  inside  surface,  which  may  become  coated  with 
scale.  An  allowance  of  i  sq.  in.  of  coil  surface  per  watt 
is  customary;  but  the  rate  at  which  heat  is  transferred 
from  the  oil  to  the  water  may  be  from  2  to  2|  times  as 
great  when  the  pipes  are  new  than  after  they  have 
become  coated  with  scale.  It  may,  therefore,  be  neces- 
sary to  clean  them  out  with  acid  at  regular  intervals,  if 
the  danger  of  high  oil  temperatures  is  to  be  avoided. 

Example.  Calculate  the  coil  surface  and  the  quan- 
tity of  water  required  for  a  transformer  with  total  losses 
amounting  to  6  kw.,  of  which  it  is  estimated  that  2 
kw.  will  be  dissipated  from  the  outside  of  the  tank. 
Surface  of  coohng  coil  =  6000  —  2000  =  4000  sq.  in. 

Assuming  a  diameter  of  ij  in.,  the  length  of  tube  in 

4000 

the  coil  will  have  to  be  — — 7^  =  85  ft. 

i2Xi.25X7r      ^ 


106  PRINCIPLES  OF  TRANSFORMER  DESIGN 

The  approximate  quantity  of  water  required  will  be 
0.25 X4  =  I  gal.  per  minute. 

30.  Transformers  Cooled  by  Forced  Oil  Circulation. 
The  transformer  and  case  are  specially  designed  so  that 
the  oil  may  be  forced  (by  means  of  an  external  pump) 
through  the  spaces  provided  between  the  coils  and  be- 
tween the  sections  of  the  iron  core.  The  ducts  may 
be  narrower  than  when  the  cooling  is  by  natural  circula- 
tion of  the  oil.  The  capacity  of  the  oil  pump  may  be 
estunated  by  allowmg  a  rate  of  flow  of  oil  through  the 
ducts  ranging  from  20  to  30  ft.  per  minute. 

It  is  not  essential  that  the  oil  be  cooled  outside  the 
transformer  case;  in  some  modem  transformers,  the  con- 
taining tank  proper  is  surrounded  by  an  outer  case,  and 
the  space  between  these  two  shells  contains  the  cooling 
coils  through  which  water  is  circulated.  These  coils, 
instead  of  being  confined  to  the  upper  portion  of  the 
transformer  case,  as  when  water  cooling  is  used  without 
forced  oil  circulation,  may  occupy  the  whole  of  the  space 
between  the  iimer  and  outer  shells  of  the  containing  tank. 
The  oil  circulation  is  obtained  by  forcing  the  oil  up 
through  the  inner  chamber  and  downward  in  the  space 
surrounding  the  water  cooling-coils. 

Such  systems  of  artificial  circulation  of  both  oil  and 
water  are  very  effective  in  connection  with  units  of  large 
output;  but  they  could  not  be  appUed  economically  to 
medium-sized  or  small  units. 


CHAPTER  IV 

MAGNETIC  LEAKAGE  IN  TRANSFORMERS— REACTANCE- 
REGULATION 

31.  Magnetic  Leakage.  Assuming  the  voltage  applied 
to  the  terminals  of  a  transformer  to  remain  constant,  it 
follows  that  the  flux  linkages  necessary  to  produce  the 
required  back  e.m.f.  can  readily  be  calculated.  The 
(vectorial)  difference  between  the  applied  volts  and  the 
induced  volts  must  always  be  exactly  equal  to  the  ohmic 
drop  of  pressure  in  the  primary  winding.  Thus,  the 
total  primary  flux  Hnkages  (which  may  include  leakage 
lines)  must  be  such  as  to  induce  a  back  e.m.f.  very  nearly 
equal  to  the  applied  e.m.f. — the  primary  IR  drop  being 
comparatively  small. 

When  the  secondary  is  open-circuited,  practically  all 
the  flux  Knking  with  the  primary  turns  Hnks  also  with 
the  secondary  turns;  but  when  the  transformer  is  loaded, 
the  m.m.f.  due  to  the  current  in  the  secondary  winding 
has  a  tendency  to  modify  the  flux  distribution,  the  action 
being  briefly  as  follows: 

The  magnetomotive  force  due  to  a  current  Is  flowing 
in  the  secondary  coils  would  have  an  immediate  effect 
on  the  flux  in  the  iron  core  if  it  were  not  for  the  fact  that 
the  slightest  tendency  to  change  the  number  of  flux  hues 
through  the  primary  coils  instantly  causes  the  primary 
current  to  rise  to  a  value  Ip  such  that  the  resultant 
107 


108  PRINCIPLES  OF  TRANSFORMER  DESIGN 

ampere  turns  {IpTp  —  hTs)  will  produce  the  exact  amount 
of  flux  required  to  develop  the  necessary  back  e.m.f. 
in  the  primary  winding.  Thus,  the  total  amount  of 
flux  linking  with  the  primary  turns  will  not  change 
appreciably  when  current  is  drawn  from  the  secondary 
terminals;  but  the  secondary  m.m.f. — together  with  an 
exactly  ^qual  but  opposite  primary  magnetizing  effect — 
will  cause  some  of  the  flux  which  previously  passed 
through  the  secondary  core  to  "  spill  over  "  and  avoid 
some,  or  all,  of  the  secondary  turns.  This  reduces  the 
secondary  volts  by  an  amount  exceeding  what  can  be 
accounted  for  by  the  ohmic  resistance  of  the  windings. 
Although  it  is  possible  to  think  of  a  leakage  field  set 
up  by  the  secondary  ampere  turns  independently  of  that 
set  up  by  the  primary  ampere  turns,  these  imaginary  flux 
components  must  be  superimposed  on  the  main  flux 
common  to  both  primary  and  secondary  in  order  that 
the  resultant  magnetic  flux  distribution  under  load  may 
be  reaUzed.  The  leakage  flux  is  caused  by  the  com- 
bined action  of  primary  and  secondary  ampere  turns, 
and  it  is  incorrect,  and  sometimes  misleading,  to  think 
Df  the  secondary  leakage  reactance  of  a  transformer  as  if 
it  were  distinct  from  primary  reactance,  and  due  to 
a  particular  set  of  flux  lines  created  by  the  secondary 
current.  In  order  to  obtain  a  physical  conception  of 
magnetic  leakage  in  transformers  it  is  much  better  to 
assume  that  the  secondary  of  an  ordinary  transformer 
has  no  5e//-inductance,  and  that  the  loss  of  pressure 
(other  than  IR  drop)  which  occurs  under  load  is  caused 
by  the  secondary  ampere  turns  diverting  a  certain  amount 
of  magnetic  flux  which,  although  it  still  links  with  the 


MAGNETIC  LEAKAGE  IN  TRANSFORMERS  109 

primary  turns,  now  follows  certain  leakage  paths  instead 
of  passing  through  the  core  under  the  secondary  coils. 

32.  Effect  of  Magnetic  Leakage  on  Voltage  Regula- 
tion. The  regulation  of  a  transformer  may  be  defined 
as  the  percentage  increase  of  secondary  terminal  voltage 
when  the  load  is  disconnected  (primary  impressed  volt- 
age and  frequency  remaining  unaltered). 

The  connection  between  magnetic  leakage  and  voltage 
regulation  will  be  studied  by  considering  the  simplest 
possible  cases,  and  noting  the  difference  in  secondary 
flux-Hnkages  under  loaded  and  open-circuited  conditions. 
The  amount  of  the  leakage  flux  in  proportion  to  the 
useful  flux  will  purposely  be  greatly  exaggerated,  and, 
in  order  to  ehminate  unessential  considerations,  the  fol- 
lowing assumptions  will  be  made : 

(i)  The  magnetizing  component  of  the  primary  cur- 
rent will  be  considered  neghgible  relatively  to  the  total 
current,  and  will  not  be  shown  in  the  diagrams. 

(2)  The  voltage  drop  due  to  ohmic  resistance  of  both 
primary  and  secondary  windings  will  be  neglected. 

(3)  The  primary  and  secondary  windings  will  be  sym- 
metrically placed  and  will  consist  of  the  same  number  of 
turns. 

(4)  One  flux  line — as  shown  in  the  diagrams — Unking 
with  one  turn  of  winding  will  generate  one  volt. 

In  Fig.  35,  both  primary  and  secondary  coils  consist 
of  one  turn  of  wire  wound  close  around  the  core:  a  cur- 
rent 7s  is  drawn  from  the  secondary  on  a  load  of  power 
factor  cos  6,  causing  a  current  /inexactly  equal  but 
opposite  to  Is — to  flow  in  the  primary  coil,  the  result  being 
the  leakage  flux  as  represented  by  the  four  dotted  lines. 


110 


PRINCIPLES  OF  TRANSFORMER  DESIGN 


The  secondary  voltage,  Es  =  2  volts,  is  due  to  the  two 
flux  lines  which  link  both  with  the  primary  and  secondary 


Fig.  35.— Magnetic  Leakage:    Thickness  of  Coils  Considered  Negligible. 

coils.  The  phase  of  this  component  of  the  total  flux  is, 
therefore,  90°  in  advance  of  Es  as  indicated  by  the  line 
OB  m  the  vector  diagram. 


MAGNETIC  LEAKAGE  IN  TRANSFORMERS  111 

In  order  to  calculate  the  necessary  primary  impressed 
e.m.f.,  we  have  as  one  component  OE'i  exactly  equal  but 
opposite  to  OEs  because  the  flux  OB  will  induce  in  the 
primary  coil  a  voltage  exactly  equal  to  Es  in  the  second- 
ary. The  other  component  is  E'iEp  =  4  volts,  equal  but 
opposite  to  the  counter  e.m.f.  which,  being  due  to  the 
four  leakage  lines  created  by  the  current  h,  will  lag  90° 
in  phase  behind  Oh.  The  resultant  is  OEp  which  scales 
5  volts.* 

When  load  is  thrown  off  the  transformer  there  will  be 
five  lines  linking  with  the  primary  which,  since  there  is 
now  no  secondary  m.m.f.  to  produce  leakage  flux,  will 
pass  through  the  iron  core  and  link  with  the  secondary. 
The  secondary  voltage  on  open  circuit  will,  therefore, 
he  Ep  =  s  volts,  and  the  percentage  regulation  is 

Ep-Es             5-2 
loox — ^ —  =  iooX =  150. 

In  Fig.  ^6,  a  departure  is  made  from  the  extreme 
simplicity  of  the  preceding  case  in  order  to  illustrate  the 
efifect  of  leakage  lines  passing  not  only  entirely  outside 
the  windings,  but  also  through  the  thickness  of  the  coils, 
as  must  always  happen  in  practical  transformers  where 
the  coils  occupy  an  appreciable  amount  of  space. 

Each  winding  now  consists  of  two  turns,  with  an  air 
space  between  the  turns  through  which  leakage  flux — 

*The  reason  why  the  six  flux  lines  shown  in  the  figure  as  linking 
with  the  primary  coil  do  not  generate  6  volts  is,  of  course,  due  to  the 
fact  that  these  flux  lines  are  not  all  in  the  same  phase;  the  resultant 
or  actual  flux  in  the  core  under  the  primary  coil  is  5  lines,  as  indicated 
by  the  vector  diagram.  The  actual  amount  of  flux  passing  any  given 
cross-section  of  the  core  must  be  thought  of  as  the  (vectorial)  addition 
of  the  flux  lines  shown  in  the  sketch  at  that  particular  section. 


112 


PRINCIPLES  OF  TRANSFORMER  DESIGN 


represented  in  Fig.  36  by  one  dotted  line — is  supposed  to 
pass.    The  single  flux  line,  linking  with  both  the  prunary 


®i 


tu 


,  ^ 

j  J    — — ^^^ 


s> 


<8) 


® 


« 


Secoi  clarjr 


8«) 


Fig.  36. — Magnetic  Leakage:  Thickness  of  Coils  Appreciable. 


and  secondary  windings,  generates  the  e.m.f.  component 
jEa  =  2  volts.    The  leakage  flux  line  marked  F  hnks  with 


MAGNETIC  LEAKAGE  IN  TRANSFORMERS  113 

only  one  turn  of  the  secondary,  and  therefore  generates 
one  volt  lagging  90°  in  phase  behind  the  primary  current 
/i.  The  total  secondary  voltage  is  Es  which  scales  2.6 
volts;  the  balancing  component  in  the  primary  being 
El.  It  should  be  particularly  noted  that  this  balancing 
component  does  not  account  for  the  full  effect  of  the  two 
flux  lines  B  and  F  linking  with  the  primary,  because, 
while  the  flux  Hne  F  links  with  only  one  secondary  turn, 
it  links  with  two  primary  turns.  The  voltage  com- 
ponent OE'i  in  the  primary  may,  therefore,  be  thought 
of  as  due  to  the  flux  hues  B  and  /,  leaving  for  the  remain- 
ing component  of  the  impressed  e.m.f.,  E'iEp  =  6  volts 
(leading  O/i  by  90°)  which  may  be  considered  as  caused 
by  the  three  Unes  F,  H,  and  G.  In  other  words,  the 
reactive  drop  {IiXp)  depends  upon  the  difference  between 
the  primary  and  secondary  flux-Hnkages  of  the  stray 
magnetic  field  set  up  by  the  combined  action  of  the 
secondary  current  Is  and  the  balancing  component  /i 
of  the  total  primary  current.  (In  this  case  /i  is  the 
total  primary  current,  since  the  magnetizing  component 
is  neglected.)  The  leakage  flux-Knkages  are  as  follows: 
With  the  primary  turns : 

7Xi  =  ivolt, 

5X2  =  2  volts, 

GX2  =  2  volts, 

FX2  =  2  volts 


7  volts 
With  the  secondary  turns: 

FXi  =  i  volt  ^ 
giving  a  difference  of  ..^Yolts^ 


114  PRINCIPLES  OF  TRANSFORMER  DESIGN 

This  is  the  vector  {hXp).    When  applying  this  rule 

T 
to  actual  transformers  in  which  the  ratio  of  turns  -^  is 

not  unity,  the  proper  correction  must  be  made  (as  ex- 
plained later)  when  calculating  the  equivalent  e.m.f. 
component  in  the  primary  circuit. 

To  obtain  the  regulation  in  the  case  of  Fig.  36,  we 
have  Ep  =  8  volts  and  Es  =  2.6  volts  when  the  transformer 
is  loaded.  When  the  load  is  thrown  off,  there  will  be 
four  flux  lines  linking  with  both  primary  and  secondary 
producing  8  volts  in  each  winding.  The  regulation  is 
therefore, 

8-2.6 
100  X — 7— =  208  per  cent. 
2.6 


33.  Experimental  Determination  of  the  Leakage  Reac- 
tance of  a  Transformer.  Although  these  articles  are 
written  from  the  viewpoint  of  the  designer,  who  must 
predetermine  the  performance  of  the  apparatus  he  is 
designing,  a  useful  purpose  will  be  served  by  considering 
how  the  leakage  reactance  of  an  actual  transformer  may 
be  determined  on  test.  The  purpose  referred  to  is  the 
clearing  up  of  any  vagueness  and  consequent  inaccuracy 
that  may  exist  in  the  mind  of  the  reader,  due  largely — 
in  the  writer's  opinion — to  the  common,  but  unnecessary 
if  not  misleading  assimiption,  that  the  secondary  has 
self  induction.* 

*  The  assumption  usually  made  in  text  books  is  that  the  secondary 
self-induction  (i.e.,  the  flux  produced  by  the  secondary  current,  and 
linking  with  the  secondary  turns)  is  equal  to  the  primary  leakage 
self-induction. 


MAGNETIC  LEAKAGE  IN  TRANSFORMERS  115 

The  diagram,  Fig.  37,  shows  the  secondary  of  a 
transformer  short-circuited  through  an  ammeter,  A, 
of  negligible  resistance.  The  impressed  primary  voltage 
Ez,  of  the  frequency  for  which  the  transformer  is  de- 
signed, is  adjusted  until  the  secondary  current  7,  is 
indicated  by  the  ammeter.  If  the  number  of  turns 
in  the  primary  and  secondary  are  Tp  and  Ts  respectively, 

the  primary  current  will  be  Ii=Isl7j^)   because,   the 

amount  of  flux  in  the  core  being  very  small,  the  mag- 


Fig.  37. — Diagram  of  Short-circuited  Transformer. 

netizing  component  of  the  primary  current  may  be 
neglected. 

The  measured  resistances  Ri  and  R2  of  the  primary 
and  secondary  coils  being  known,  the  vector  diagram 
Fig.  38,  can  be  constructed. 

The  volts  induced  in  the  secondary  are  OE2  (equal 
to  IsR2)  in  phase  with  the  current  L.  The  bal- 
ancing   component    in    the   primary   winding   is   0E\ 

equal  to  £2(  -^  )    in  phase  with  the  primary  current  /i. 

Another  component  in  phase  with  this  current  is  E\P 
(equal   to   IiRi).     Since   the   total  impressed   voltage 


116 


PRINCIPLES  OF  TRANSFORMER  DESIGN 


has  the  known  value  E^,  we  can  describe  an  arc  of 
circle  of  radius  OE^  from  the  point  O  as  a  center.  By 
erecting  a  perpendicular  to  01  \  at  the  point  P,  the 
point  E,  is  determined,  and  E^P  is  the  loss  of  pressure 
caused  by  magnetic  leakage.  The  vector  OP  may  be 
thought  of  as  the  product  of  the  primary  current  h, 


Fig.  38. — Vector  Diagram  of  Short-circuited  Transformer. 

and  an  equivalent  primary  resistance  Rp,  which  assumes 
the  secondary  resistance  to  be  zero,  but  the  primary 
resistance  to  be  increased  by  an  amount  equivalent 
to  the  actual  secondary  resistance.     Thus, 


I\Rj,  =  IiRi  -\-IsR2 


TsP 


but 


L=I 


®- 


MAGNETIC  LEAKAGE  IN  TRANSFORMERS  117 

whence, 

R,  =  Ri+R2(^y (35) 

In  order  to  get  an  expression  for  the  transformer 
leakage  reactance  (Xp)  in  terms  of  the  test  data,  we 
can  write, 

whence 


1^2 


This  quantity,  multiplied  by  h  (or  IiXp  = 
VEJ^-ihRpf)  is  the  vector  E\Ej,  of  the  diagrams 
in  Figs.  35  and  36  as  it  might  be  determined  experi- 
mentally for  an  actual  transformer.  If  it  were  possible 
for  all  the  magnetic  flux  to  Unk  with  all  the  primary, 
and  all  the  secondary,  turns,  the  quantity  hXp  would 
necessarily  be  zero;  all  the  flux  would  be  in  the  phase 
OB,  and  OEz  (of  Fig.  38)  would  be  equal  to  OP.  The 
presence  of  the  quantity  hXp  can  only  be  due  to  those 
flux  hnes  which  link  with  primary  turns,  but  do  not  link 
with  an  equivalent  number  of  secondary  turns. 

34.  Calculation  of  Reactive  Voltage  Drop.  Seeing  that 
it  is  generally — although  not  always — desirable  to  obtain 
good  regulation  in  transformers,  it  is  obvious  that  designs 
with  the  primary  and  secondary  windings  on  separate 
cores  (see  Figs,  i,  35  and  36),  which  greatly  exaggerate 
the  ratio  of  leakage  flux  to  useful  flux,  would  be  very 
unsatisfactory  in  practice.  By  putting  half  the  primary 
and  half  the  secondary  on  each  of  the  two  limbs  of  a 


118 


PRINCIPLES  OF  TRANSFORMER  DESIGN 


single-phase  core-type  transformer,  as  shown  in  Fig.  39, 
a  considerable  improvement  is  effected,  but  the  reluctance 
of  the  leakage  paths  is  still  low,  and  this  design  is  not 
nearly  so  good  as  Fig.  7  (page  18)  where  the  leakage  paths 
have  a  greater  length  in  proportion  to  the  cross-section. 
Similarly  in  the  shell  type  of  transformer,  the  design 
shown  in  Fig.  40  is  unsatisfactory;   the  arrangement  of 


Fig.  39. — Leakage  Flux  Lines  in  Special  Core-type  Transformer. 


coils,  as  shown  in  Figs.  10  and  11  (Art.  8)  is  much  better 
because  of  the  greater  reluctance  of  the  leakage  paths. 
Transformers  with  coils  arranged  as  in  Figs.  7  and  10  are 
satisfactory  for  small  sizes;  but,  in  large  units,  it  is  neces- 
sary to  subdivide  the  windings  into  a  large  number  of 
sections  with  primary  coils  "  sandwiched "  between 
secondary  coils  as  in  Fig.  1 7  (core  type)  and  Figs.  8  and 


MAGNETIC  LEAKAGE  IN  TRANSFORMERS 


119 


1 6  (shell  type).  By  subdividing  the  windings  in  this 
manner,  the  m.m.f.  producing  the  leakage  flux,  and  the 
number  of  turns  which  this  flux  hnks  with,  are  both 
greatly  reduced.  The  objection  to  a  very  large  number 
of  sections  is  the  extra  space  taken  up  by  insulation 
between  the  primary  and  secondary  coils.  For  the 
purpose  of  facilitating  calculations,  the  windings  of 
transformers  can  generally  be  divided  into  unit  sections 


Fig.  40.— Leakage  Flux  Lines  in  Poorly  Designed  Shell-type  Transformer. 


as  indicated  in  Fig.  41  (which  shows  an  arrangement  of 
coils  in  a  shell- type  transformer  similar  to  Fig.  16). 
Each  section  consists  of  half  a  primary  coil  and  half  a 
secondary  coil,  with  leakage  flux  passing  through  the 
coils  and  the  insulation  between  them  *  all  in  the  same 

*  If  air  ducts  are  required  between  sections  of  the  winding,  these  should 
be  provided  in  the  position  of  the  dotted  center  lines,  by  a  further  sub- 
division of  each  primary  and  secondary  group  of  turns;  thus  allowing 
the  space  between  primary  and  secondary  coils  to  be  filled  with  solid 
insulation.  It  is  evident  that,  if  good  regulation  is  desired,  the  space 
between  primary  and  secondary  coils— where  the  leakage  flux  density 
has  its  maximum  value — must  be  kept  as  small  as  possible. 


120  PRINCIPLES  OF  TRANSFORMER  DESIGN 

direction,  as  indicated  by  the  flux  diagram  at  the  bottom 
of  the  figure. 


Fig.  41.— Section  through  Coils  of  Shell-type  Transformer. 

The  effect  of  all  leakage  lines  in  the  gap  between  the 
coils  is  to  produce  a  back  e.m.f.  in  the  primary  without 


MAGNETIC  LEAKAGE  IN  TRANSFORMERS  121 

affecting  the  voltage  induced  in  the  secondary  by  the 
main  component  of  the  total  flux  (represented  by  the 
full  line).  Of  the  other  leakage  lines,  B  hnks  with  only 
a  portion  of  the  primary  turns  and  has  no  effect  on  the 
primary  turns  which  it  does  not  Hnk  with ;  while  A  links 
not  only  with  all  the  primary  turns,  but  also  with  a  cer- 
tain number  of  secondary  turns.  Note  that  if  the  line  A 
were  to  coincide  with  the  dotted  center  Hne  MN, 
marking  the  limit  of  the  unit  section  under  consideration, 
it  would  have  no  effect  on  the  transformer  regulation 
because  flux  which  links  equally  with  primary  and 
secondary  is  not  leakage  flux.  Actually,  the  hne  A 
links  with  all  the  primary  turns  of  the  half  coil  in  the 
section  considered,  but  with  only  a  portion  of  the  second- 
ary turns  in  the  same  section.  Its  effect  is,  therefore, 
exactly  as  if  it  Hnked  with  only  a  fractional  number  of 
the  primary  turns.  The  mathematical  development 
which  follows  is  based  on  these  considerations. 

Fig.  42  is  an  enlarged  view  of  the  unit  section  of  Fig.  41, 
the  length  of  which — measured  perpendicularly  to  the 
cross  section — is  I  cms.  All  the  leakage  is  supposed  to 
be  along  parallel  lines  perpendicular  to  the  surface  of  the 
iron  core  above  and  below  the  coils. 

It  is  desired  to  calculate  the  reactive  voltage  drop  in  a 
section  of  the  winding  of  length  I  cms.,  depth  k  cms.,  and 
total  width  {s-\-g-{-p)  cms.,  where 

5  =  the  half  thickness  of  the  secondary  coil; 

g  =  the  thickness  of   insulation   between  primary  and 

secondary  coils; 
p  =  the  half  thickness  of  the  primary  coil. 


122 


PRINCIPLES  OF  TRANSFORMER  DESIGN 


The  voltage  drop  caused  by  the  leakage  flux  in  the 
spaces  g,  p,  and  s  will  be  calculated  separately  and  then 
added  together  to  obtain  the  total  reactive  voltage  drop. 
The  general  formula  giving  the  r.m.s.  value  of  the  volts 
induced  by  $  maxwells  linking  with  T  turns  is 


/X=-^/$rxio-8, 

V2 


(36) 


when  the  flux  variation  follows  the  simple  harmonic  law. 


Fig.  42. — Enlarged  Section  through  Transformer  Coils. 

In  calculating  the  voltage  produced  by  a  portion  of 
flux  in  a  given  path,  we  must  therefore  determine  (i) 


MAGNETIC  LEAKAGE  IN  TRANSFORMERS  123 

the  amount  of  this  flux,  and  (2)  the  number  of  turns 
with  which  it  links.  The  symbols  Ti  and  T2  will  be 
used  to  denote  the  number  of  turns  in  the  half  sections 
of  widths  p  and  5  of  the  primary  and  secondary  coils 
respectively.  The  meaning  of  the  variables  x  and  y 
is  indicated  in  Fig.  42.     The  symbol  m  will  be  used  for 

the  quantity  — 4- 

For  the  section  g  we  have, 

V2 

Inserting  for  4>  its  value  in  terms  of  m.m.f.  and  per- 
meance, this  becomes, 


{IX),  =  m{o.AirTih)x'jXT,.       .     .     (37) 


In  the  section  p,  the  m.m.f.  producing  the  element 
of  flux  in  the  space  of  width  dx  is  due  to  the  current 

1 1  in  (-)^i  turns,  and  since  this  element  of  flux  links 
with  only  ( - )  ^1  turns,  we  have, 

d(IX),  =  n.[o.4.(fjT.l]'fx(f)Tu 

whence 

{IX).  =  mX^^^'^rx^dx 
P^h      Jo 

=  mXo.4TTiHM (38) 

3/? 


124  PRINCIPLES  OF  TRANSFORMER  DESIGN 

In  the  section  s,  the  m.m.f., producing  the  small  element 
of  flux  in  the  space  of  width  dy  is  due  to  the  current 

/,  in  1-)Ts  turns,  and  since  this  must  be  considered 
as  hnking  with  f  i j  Tx  turns,  we  can  write, 

d(/x),=m[o.4.(5r./.]«^Qr, 

whence 

=  mXo.4TrTrIi—, (39) 

wherefrom  the  secondary  quantities  T2  and  /«  have  been 
eliminated  by  putting  (TJi)  in  place  of  {T2ls). 

The  final  expression  for  the  inductive  voltage  drop 
in  the  unit  section  considered  is  obtained  by  adding 
together  the  quantities  (37),  (38),  and  (39).     Thus, 

10**/^  L         3    J 

wherein  all  dimensions  are  expressed  in  centimeters. 
If  all  the  primary  turns  are  connected  m  series,  this 

T 

quantity  will  have  to  be  multiplied  by  the  ratio  ^  to 

obtain  the  value  of  the  vector  h  Xj,  shown  in  the  vector 
diagrams. 


MAGNETIC  LEAKAGE  IN  TRANSFORMERS  125 

Equivalent  Value  of  the  Length  I.  The  numerical  value 
of  the  length  I  as  used  in  the  above  formulas  might 
reasonably  be  taken  as  the  mean  length  per  turn  of  the 
transformer  windings,  provided  the  reluctance  of  the 
flux  paths  outside  the  section  shown  in  Fig.  42  may  be 
neglected,  not  only  where  the  iron  laminations  provide 
an  easy  path  for  the  flux,  but  also  where  the  ends  of  the 
coils  project  beyond  the  stampings. 

Every  manufacturer  of  transformers  who  has  accu- 
mulated sufficient  test  data  from  transformers  built 
to  his  particular  designs,  will  be  in  a  position  to  modify 
Foi-mula  (40)  in  order  that  it  may  accord  very  closely 
with  the  measured  reactive  voltage  drop.  This  correc- 
tion may  be  in  the  form  of  an  expression  for  the  equiva- 
lent length  /,  which  takes  into  account  the  type  of 
transformer  (whether  core  or  shell)  and  the  arrange- 
ment  of   coils;     or    the   quantity    \g-\-- —      may   be 

modified,  being  perhaps  more  nearly  \g-\-- —    ,  which 

allows  for  more  leakage  flux  through  the  space  occupied 
by  the  copper  than  is  accounted  for  on  the  assumption 
of  parallel  flux  lines.  The  writer  believes,  however, 
that  if  /  is  taken  equal  to  the  mean  length  per  turn 
of  the  windings — expressed  in  centimeters — the  Formula 
(40)  will  yield  results  sufficiently  accurate  for  nearly 
all  practical  purposes. 

35.  Calculation  of  Exciting  Current.  Before  drawing 
the  complete  transformer  vector  diagram,  including  the 
reactive  drop  calculated  by  means  of  the  formula  devel- 
oped in  the  preceding  article,  it  is  necessary  to  consider 


126 


PRINCIPLES  OF  TRANSFORMER  DESIGN 


how  the  magnitude  and  phase  of  the  exciting  current 
component  of  the  total  primary  current  may  be  pre- 
determined. 

The  exciting  current  (/«)  may  be  thought  of  as  con- 
sisting of  two  components:  (i)  the  magnetizing  com- 
ponent (/o)  in  phase  with  the  main  component  of  the 
magnetic  flux,  i.e.,  that  which  links  with  both  primary 
and  secondary  coils,  and  (2)  the  "  energy  "  component 


3 

h^ 

I„ 

^^"1 

f         Sf AX.  valne  of  carrent  component 

\'2- 

!     \ 

Amp.  turns  to  produoeB  max. 

V7T, 

E'l 

K,  ^^-x— ^< 

3 

PluMe  of  Indnoed    e.  m.  /. 

Fig.  43. 


.  Total  iron  loa«  (watts) 

'"  Primary  impressed  voltn. 

-Vector  Diagram  showing  Components  of  Exciting  Current. 


[Iv>)  leading  7o  by  one-quarter  period,  and,  therefore, 
exactly  opposite  in  phase  to  the  induced  e.m.f. 

The  magnitude  of  this  component  depends  upon  the 
amount  of  the  iron  losses  only,  because  the  very  small 
copper  losses  {PeRi)  may  be  neglected. 

If  these  components  could  be  considered  sine  waves, 
the  vector  construction  of  Fig.  43  would  give  correctly 
the  magnitude  and  phase  of  the  total  exciting  current  L. 
For  values  of  flux  density  above  the  "  knee  "  of  the 
B-  H  curve,  the  instantaneous  values  of  the  magnetizing 
current  are  no  longer  proportional  to  the  flux,  and  this 


MAGNETIC  LEAKAGE  IN  TRANSFORMERS  127 

component  of  the  total  exciting  current  cannot  therefore 
be  regarded  as  a  sine  wave  even  if  the  flux  variations 
are  sinusoidal.  The  error  introduced  by  using  the  con- 
struction of  Fig.  43  is,  however,  usually  negligible  because 
the  exciting  current  is  a  very  small  fraction  of  the  total 
primary  current. 

The  notes  on  Fig.  43  are  self  explanatory,  but  reference 
should  be  made  to  Fig.  44  from  which  the  ampere  turns 
per  inch  of  the  iron  core  may  be  read  for  any  value  of 
the  (maximum)  flux  density.  The  flux  density  is  given 
in  gausses,  or  maxwells  per  square  centimeter  of  cross- 
section.*  The  total  magnetizing  ampere  turns  are  equal 
to  the  number  read  off  the  curve  multiplied  by  the  mean 
length  of  path  of  the  flux  which  links  with  both  primary 
and  secondary  coils.  When  butt  joints  are  present  in 
the  core,  the  added  reluctance  should  be  allowed  for. 
Each  butt  joint  may  be  considered  as  an  air  gap  0.003  in. 
long,  and  the  ampere  turns  to  be  allowed  in  addition  to 
those  for  the  iron  portion  of  the  magnetic  circuit  are 
therefore, 

Amp.  turns  Jor  joints  = 

0.47r 

=0.006  X-Smax  XNo.  of  butt  joints  in  series.     (41) 

Instead  of  calculating  the  exciting  current  by  the 
method  outhned  above,  designers  sometimes  make  use 

*  The  writer  makes  no  apology  for  using  both  the  inch  and  the  centi- 
meter as  units  of  length.  So  long  as  engineers  insist  that  the  inch  has 
certain  inherent  virtues  which  the  centimeter  does  not  possess,  they 
should  submit  without  protest  to  the  inconvenience  and  possible  dis- 
advantage of  having  to  use  conversion  factors,  especially  in  connection 
with  work  based  on  the  fundamental  laws  of  physics. 


128  PRINCIPLES  OF  TRANSFORMER  DESIGN 


16000 

^ 

15000 

^ 

•i> 

^ 

iN 

^ 

.A 

f 

f^ 

13000 

/ 

/ 

/ 

/ 

^12000 

/ 

/ 

Si  1000 

/ 

/ 

§10000 

/ 

/ 

9000 

/ 

' 

8000 

7000 

6000 

5000 

0  10  20  30  40  50  60  70  80 

Ampere-turns  per  inch 

Fig.  44. — Curve  giving  Connection  between  Magnetizing  Ampere-turns 
and  Flux  Density  in  Transformer  Iron. 


MAGNETIC  LEAKAGE  IN  TRANSFORMERS  129 

of  curves  connecting  maximum  core  density  and  volt- 
amperes  of  total  exciting  current  per  cubic  inch  or  per 
pound  of  core;  the  data  being  obtained  from  tests  on 
completed  transformers.  The  fact  that  the  total  volt- 
amperes  of  excitation  (neglecting  air  gaps)  are  some 
function  of  the  flux  density  multiplied  by  the  weight  of 
the  iron  in  the  transformer  core,  may  be  explained  as 
follows : 

Let  w  =  total  watts  lost  per  pound  of  iron,  correspond- 
ing to  a  particular  value  of  B  as  read  off 
one  of  the  curves  of  Fig.  27; 
a = Ampere  turns  per  inch  as  read  off  Fig.  44; 
A  =  cross-section   of  iron  in  the  core,  measured 
perpendicularly  to  the  magnetic  flux  hues 
(square  inches) ; 
/  =  Length  of  the  core  in  the  direction  of  the 

flux  Hues  (inches) ; 
P  =  Weight  of  core  in  pounds  =  0.28.4/. 

The  symbols  previously  used  are: 
Tp  =  number  of  primary  turns : 
£,  =  primary  ^at,^'^-44TJM5BA)f^ 

Given  definite  values  for  B  and  /,  the  "  in  phase  " 
component  of  the  exciting  current  is 

,  _core  loss_'Z£;XP 
Ep  Ep 


130  PRINCIPLES  OF  TRANSFORMER  DESIGN 

component,  or  true  magnetizing 


and 

the  " 

wattless  ' 

'  con 

current,  is 

/o  = 

aXl 

whence 


'-JSNS"- 


p 

Multiplying  both  sides  of  the  equation  by  -^,  we  get 

£p/e  _  volt-amperes  of  total  excitation 
P  weight  of  core 


■■<^^H 


This  formula  may  be  used  for  plotting  curves  such 
as  those  in  Fig.  45.     Thus,  if 

5  =  13,000  gausses, 

/=6o  cycles  per  second, 
w;  =  i.55  (read  off  curve  for  silicon  steel  in  Fig.  27), 

a  =  22  (from  Fig.  44);  and,  by  Formula  (42) 

Volt-amperes  per  pound 


=  V(i.55)^+( 


4.44  X  22  X  13,000  X  6.45  X6o\2 

0.28X108  '  ~'^'^- 


The  error  in  this  method  of  deriving  the  curves  of 
Fig.  45  is  due  to  the  fact  that  sine  waves  are  assumed. 
The  data  for  plotting  the  curves  should  properly  be 
obtained  from  tests  on  cores  made  out  of  the  material 
to  be  used  in  the  construction  of  the  transformer. 


MAGNETIC  LEAKAGE  IN  TRANSFORMERS  131 


16000 

^ 

15000 

^ 

f>  ^ 

^ 

^ 

-^ 

14000 

"y 

X 

-^ 

^ 

^ 

" 

/ 

^ 
^ 

■^ 

■^  nnnn 

/ 

P 

y' 

^ 

/ 

/ 

<^ 

A 

;^i2ooo 

/ 

/ 

/ 

'/ 

/ 

// 

^ 11000 

/ 

/ 

/ 

/ 

value  of 

1 

y 

V 

/ 

1 

0 

2    Qfinn 

' 

// 

// 

8000 

I 

'/ 

/ 

/ 

7000 

[ 

1 

'( 

5000 

10  15  20  25  30  35 

Exciting  volt-amperes  per  lb.  of  stampings 
(Approximate  values  for  either  iron  or  silicon-steel) 

Fig.  45.— Curv^es  giving  Connection  between   Exciting   Volt-amperes 
and  Flux  Density  in  Transformer  Stampings. 


132  PRINCIPLES  OF  TRANSFORMER  DESIGN 

The  eflfect  of  the  magnetizing  current  component  in 
distorting  the  current  waves  may  be  appreciable  when 
the  core  density  is  carried  up  to  high  values.  The  curve 
of  flux  variation  cannot  then  be  a  sine  wave,  and  the 
introduction  of  high  harmonics  in  the  current  wave  may 
aggravate  the  disturbances  that  are  always  hable  to 
occur  in  telephone  circuits  paralleling  overhead  trans- 
mission lines.  This  is  one  reason  why  high  values  of 
the  exciting  current  are  objectionable.  An  open-circuit 
primary  current  exceeding  lo  per  cent  of  the  full-load 
current  would  rarely  be  permissible. 

36.  Vector  Diagrams  Showing  Effect  of  Magnetic 
Leakage  on  Voltage  Regulation  of  Transformers.  The 
vector  diagrams,  Figs.  46,  47,  and  48,  have  been  drawn 
to  show  the  voltage  relations  in  transformers  having 
appreciable  magnetic  leakage.  The  proportionate  length 
of  the  vectors  representing  IR  drop.  IX  drop,  and 
magnetizing  current,  has  purposely  been  exaggerated  in 
order  that  the  construction  of  the  diagrams  may  be 
easily  followed. 

Fig.  46  is  the  complete  vector  diagram  of  a  transformer; 
the  meaning  of  the  various  component  quantities  being 
as  follows: 

£2  =  Induced  secondary  e.m.f.,  due  to  the  flux  {OB) 
linking  with  the  secondary  turns; 

Es  =  Secondary  terminal  voltage  when  the  secondary 
current  is  L  amperes  on  a  load  power  factor 
of  cos  d; 

/e  =  Primary  exciting  current,  calculated  as  ex- 
plained in  the  preceding  article; 


MAGNETIC  LEAKAGE  IN  TRANSFORMERS  133 

7i  =  Balancing  component  of  total  primary  current 


(='■4:): 


/p  =  Total  primary  current; 
£'i  =  Balancing  component  of  induced  primary  volt- 


age 


{-^^^'W' 


PE'i=IR   drop   due    to   primary   resistance    (drawn 

parallel  to  01  p) ; 
EpP  =  IX  drop  due  to  leakage  reactance  (drawn  at 

right  angles  to  01  p) ; 
Ep  =  Impressed  primary  e.m.f . 


Fig.  46. — Vector  Diagram  of  Transformer  on  Inductive  Load. 

It  is  usually  permissible  to  neglect  the  exciting  current 
component  when  considering  full-load  conditions.  This 
leads  to  the  simpler  diagram,  Fig.  47,  in  which  the  total 
primary  current  is  supposed  to  be  of  the  same  magnitude 
and  phase  as  what  has  previously  been  referred  to  as  the 
balancing  component  of  the  total  primary  current. 

The  dotted  hues  in  Fig.  47  show  how  a  still  greater 
simpHfication   may   be   effected   in   drawing   a   vector 


134 


PRINCIPLES  OF  TRANSFORMER  DESIGN 


diagram  from  which  the  voltage  regulation  can  be  cal- 
culated. Instead  of  drawing  the  two  vectors  OE2 
and  OEs  for  the  induced  and  terminal  secondary  voltages, 
we  can  draw  OE^  opposite  in  phase  to  £j  and  equal  to 

e/^V     Then  EeP   (drawn   parallel    to  Oh)    is   the 

component  of  the  impressed  primary  volts  necessary 
to  overcome  the  ohmic  resistance  of  both  primary  and 
secondary  windings. 


Fig.  47. — Simplified  Vector  Diagram  of  Transformer;  Exciting  Current 
Neglected. 


It  is  now  only  necessary  to  turn  this  diagram  through 
180  degrees,  and  eliminate  all  unnecessary  vectors,  in 
order  to  arrive  at  the  very  simple  diagram  of  Fig.  48, 
from  which  the  voltage  regulation  can  be  calculated. 

37.  Formulas    for    Voltage    Regulation.      From    an 
inspection  of  Fig.  48,  it  is  seen  that 


Ep- 


{IlRp)-\-EeCOSd 

cos  <t>         ' 


.     .     (43) 


MAGNETIC  LEAKAGE  IN  TRANSFORMERS  135 

wherein  cos  6  is  known  (being  the  power  factor  of  the 
external  load),  and  cos  4>  has  not  yet  been  determined. 
But, 


tan  (j)- 


{hXp)  -\-Ee  sin  d 

{IlRj>)-{-EeCOSd' 


(44) 


Fig.   48. — Simple   Transformer   Vector   Diagram   for   Calculation  of 
Voltage  Regulation. 


which  can  be  used  to  calculate  </>  and  therefore  cos  <t>. 
The  percentage  regulation  is 


100  X—^^:^ —  =  iooX ^ -■ ^,     (45) 

Ee  Ee  cos  0  '     ^^^^ 


or,  if  the  ohmic  drop  is  expressed  as  a  percentage  of  the 
(lower)  terminal  voltage: 
Per  cent  regulation 


Per  cent  equiv.  IR  drop+ioo(cos  0— cos  0) 
cos  0 


.     (46) 


136  PRINCIPLES  OF  TRANSFORMER  DESIGN 

The  difference  between  the  angles  6  and  <f>  (Fig. 
48)  is  generally  small,  and  it  is  then  permissible  to 
assume  that  OD  =  OE,.    But 

OD  =  Ee-\-IiRu  cos  d-hIiXi,sin  6, 
whence, 

Per  cent  regulation  (approximate) 

=  Per  cent  IR  cos  0+per  cent  IX  sin  d.     (47) 

If  the  power  factor  were  leading  instead  of  lagging  as 
in  Fig.  48,  the  plus  sign  would  have  to  be  changed  to  a 
minus  sign. 

Example.  In  order  to  show  that  the  approximate 
Formula  (47)  is  sufficiently  accurate  for  practical  pur- 
poses, the  following  numerical  values  are  assumed. 

Power  factor  (cos  6)  =0.8. 
Total  IR  drop  =  1.5  per  cent. 
Total  IX  drop  =  6.0  per  cent. 

By  Formula  (44), 

0.06+0.6        o 

tan  (t>  = — —  =0.81, 

0.015+0.8 

whence  cos  0  =  0.777,  and,  by  Formula  (46), 

Regulation  =  ^'^      ^"^^''^=4.9  per  cent. 

By  the  approximate  Formula  (47), 

Regulation  =  (i.5Xo.8)  +  (6Xo.6)=4.8  per  cent. 


MAGNETIC  LEAKAGE  IN  TRANSFORMERS  137 

The  total  equivalent  voltage  drop,  due  to  the  resistance 
of  the  windings  (the  quantity  I\Rp  of  the  vector  dia- 
grams) is  usually  between  i  and  2  per  cent  of  the  ter- 
minal voltage  in  modern  transformers.  The  reactive 
voltage  drop  caused  by  magnetic  leakage  (the  quantit}^ 
IiXp  in  the  vector  diagrams)  is  nearly  always  greater 
than  the  IR  drop,  being  3  to  8  per  cent  of  the  terminal 
voltage.  Sometimes  it  is  10  per  cent,  or  even  more, 
especially  in  high- voltage  transformers  where  the  space 
occupied  by  insulation  is  considerable,  or  in  transformers 
of  very  large  size,  when  the  object  is  to  keep  the  current 
on  short  circuit  within  safe  limits. 


CHAPTER  V 

PROCEDURE  IN  TRANSFORMER  DESIGN 

37.  The  Output  Equation.  The  volt-ampere  output 
of  a  single-phase  transformer  is  £X/  which,  as  explained 
in  Art.  6,  may  be  written 

Volt-amperes  =  ^^^X*X(r/),     .     .     (48) 

where  TI  stands  for  the  total  ampere  turns  of  either  the 
primary  or  secondary  winding. 

There  is  no  limit  to  the  number  of  designs  which  will 
satisfy  this  equation;  the  total  flux,  4>,  is  roughly  a 
measure  of  the  cross-section  of  the  iron  core,  while  the 
quantity  {TI)  determines  the  cross-section  of  the  wind- 
ings. The  problem  before  the  designer  is  to  proportion 
the  parts  and  dispose  the  material  in  such  a  way  as  to 
obtain  the  desired  output  and  specified  efficiency  at  the 
lowest  cost.  The  temperature  rise  is  also  a  matter  of 
importance  which  must  be  watched,  and  light  weight 
is  occasionally  more  important  than  cost. 

It  cannot  be  said  that  there  is  one  method  of  attacking 
the  problems  of  transformer  design  which  has  indisputable 
advantages  over  all  others;  and  in  this,  as  in  all  design, 
the  judgment  and  experience  of  the  individual  designer 
must  necessarily  play  an  important  part.  The  apparent 
138 


PROCEDURE  IN  TRANSFORMER  DESIGN  139 

simplicity  of  the  calculations  involved  in  transformer 
design  is  the  probable  cause  of  the  many  more  or  less 
unsuccessful  attempts  to  reach  the  desired  end  by  purely 
mathematical  methods.  It  is  not  possible  to  include 
all  the  variable  factors  in  practical  mathematical  equa- 
tions purporting  to  give  the  ideal  quantities  and  pro- 
portions to  satisfy  the  specification.  Methods  of  pro- 
cedure aiming  to  dispense  with  individual  judgment  and 
a  certain  amount  of  correction  or  adjustment  in  the  final 
design,  should  generally  be  discountenanced,  because 
they  are  based  on  inadequate  or  incorrect  assumptions 
which  are  Hable  to  be  overlooked  as  the  work  proceeds 
and  becomes  finally  crystaUized  into  more  or  less  for- 
midable equations  and  formulas  of  unwieldy  propor- 
tions. 

No  claim  to  originality  is  made  in  connection  with 
the  following  method  of  procedure;  indeed  it  is  ques- 
tionable whether  the  mass  of  existing  literature  treating 
of  the  alternating  current  transformer  leaves  anything 
new  to  be  said  on  the  subject  of  procedure  in  design. 
All  that  the  present  writer  hopes  to  present  is  a  treat- 
ment consistent  with  what  has  gone  before,  based  always 
on  the  fundamental  principles  of  physics — even  though 
the  use  of  empirical  constants  may  be  necessary. 

Instead  of  attempting  to  take  account  at  one  time  of 
all  the  conditions  to  be  satisfied  in  the  final  design,  the 
factors  which  have  the  grei.test  influence  on  the  dimen- 
sions will  be  considered  first;  items  such  as  temperature 
rise  and  voltage  regulation  being  checked  later  and,  if 
necessary,  corrected  by  slight  changes  in  the  dimensions" 
or  proportions  of  the  preliminary  design. 


140  PRINCIPLES  OF  TRANSFORMER  DESIGN 

38.  Specifications.  It  will  be  advisable  to  list  here 
the  particulars  usually  specified  by  the  buyer,  and  sup- 
plement these,  if  required,  with  certain  assumptions  that 
the  manufacturer  must  make  before  he  can  proceed 
with  a  particular  design. 

(i)  K.v.a.  output. 

(2)  Number  of  phases. 

(3)  Primary  and  secondary  voltages  {Ep  and  £,). 

(4)  Frequency  (/). 

(5)  Efficiency  under  specified  conditions. 

(6)  Voltage  regulation  under  specified  load. 

(7)  Method  of  cooHng — Temperature  rise. 

(8)  Maximum  permissible  open-circuit  exciting  cur- 
rent. 

Items  (i)  to  (4)  must  always  be  stated  by  the  pur- 
chaser, while  the  other  items  may  be  determined  by  the 
manufacturer,  who  should,  however,  be  called  upon  to 
furnish  these  particulars  in  connection  with  any  competi- 
tive oflfer. 

With  reference  to  item  (5),  if  the  efficiency  is  stated 
for  two  different  loads,  the  permissible  copper  and  iron 
losses  can  be  calculated.  If  the  buyer  does  not  furnish 
these  particulars,  he  should  state  whether  the  trans- 
former is  for  use  in  power  stations  or  on  distributing 
lines ,  in  order  that  the  relation  of  the  iron  losses  to  the 
total  losses  may  be  adjusted  to  give  a  reasonable  all-day 
efficiency.  In  any  case,  before  proceeding  with  the 
design,  the  maximum  permissible  iron  and  copper  losses 
must  be  known  or  assumed. 

The  requirements  of  items  (6),  (7),  and  (8),  are  to  some 


PROCEDURE  IN  TRANSFORMER  DESIGN         (J^ 

extent  satisfied,  even  in  the  preliminary  design,  by- 
selecting  a  flux  density  (B)  and  a  current  density  (A) 
from  the  values  given  in  Article  20,  because  industrial 
competition  and  experience  have  shown  these  values 
to  give  the  best  results  while  using  the  smallest  per- 
missible amount  of  material.  Thus,  by  selecting  a 
proper  value  for  A,  both  the  local  heating  and  the  IR 
drop  of  the  windings  will  probably  be  within  reason- 
able hmits.  The  other  factor  influencing  the  voltage 
regulation  (item  (6))  is  the  reactive  drop,  which  can 
generally  be  controlled  by  suitably  subdividing  the 
windings. 

A  proper  value  of  the  flux  density  (B)  will  generally 
keep  item  (8)  within  the  customary  hmits. 

39.  Estimate  of  Number  of  Turns  in  Windings.  Re- 
turning to  the  Formula  (48)  in  Article  37,  if  a  suitable 
value  for  T  could  be  determined  or  assumed,  the  only 
unknown  quantity  in  the  output  equation  would  be  $ 
and  we  should  then  have  a  starting-point  from  which  the 
dimensions  of  a  preliminary  design  could  be  easily  cal- 
culated. 

■  Let  F«  =  volts  per  turn  (of  either  primary  or  sec- 
ondary winding)  then,  in  order  to  express  this  quan- 
tity in  terms  of  the  volt-ampere  output,  we  have, 


£_(£/) 
'    T      TI' 


from  which  T  must  be  eliminated,  since  the  reason 
for  seeking  a  value  for  Vt  is  that  T  may  be  calculated 
therefrom. 


142  PRINCIPLES  OF  TRANSFORMER  DESIGN 

Using  the  value  of  {EI)  as  given  by  Formula  (48), 
we  can  write 


£/     444/4>r/ 
Vt  -xi^TIXio^' 


whence 


F,  =  Vvolt-ampere  output  X^'^-^^'^/^V        (49) 

The  quantity  in  brackets  under  the  second  radical  is 
found  to  have  an  approximately  constant  value,  for  an 
efficient  and  economical  design  of  a  given  type,  without 
reference  to  the  output.  This  permits  of  the  formula 
being  put  in  the  form 


Vt  =  cX  Vvolt-ampere  output,      ....  (49a) 


where  c  is  an  empirical  coefficient  based  on  data  taken 
from  practical  designs. 

Factors    Influencing    the    Value   of  the   Coefficient   c. 

f^ 
It  is  proposed  to  examine  the  meaning  of  the  ratio  ~ 

which  appears  under  the  second  radical  of  Formula  (49) 
with  a  view  to  expressing  this  in  terms  of  known  quan- 
tities, or  of  quantities  that  can  easily  be  estimated. 
Let  Wc  =  full  load  copper  losses  (watts) ; 
Wi  =  core  losses  (watts) ; 
the  relation  between  these  losses  being; 

Wc  =  bWu (so) 


PROCEDURE  IN  TRANSFORMER  DESIGN  143 

wherein  b  must   always  be  known  before  proceeding 
with  the  design. 

Let  /c  =  mean  length  per  turn  of  copper  in  windings; 
/i  =  mean   length    of   magnetic    circuit   measured 
along  flux  hnes; 
then  /  i!^uu^4i /<  -  -'o 

Wc  =  constant  X  A^  Xvolume  of  copper 

=  kc{TI)Mc, (51) 

where  h  is  a  constant  to  be  determined  later.     Similarly 

Wi  =  constant  X/B"  Xvolume  of  iron 


=  kJB^{l) 


=h{mB^-% (52) 

wherein  ki  is  another  constant  to  be  determined  later. 

Inserting  these  values  in  Formula  (50),  the  required 
ratio  can  be  put  in  the  form 

f^^_kcA_/lc\  ,      . 

TI    hk^B''-\lJ ^^^^ 

This  ratio  is  thus  seen  to  depend  on  certain  quantities 
and  constants  which  are  only  slightly  influenced  by  the 
output  of  the  transformer.  They  depend  on  such  items 
as  the  ratio  of  copper  losses  to  iron  losses  (i.e.,  whether 
the  transformer  is  for  use  on  power  transmission  Hnes,  or 
distributing  circuits) ;    temperature  rise  and  methods  of 


144  PRINCIPLES  OF  TRANSFORMER  DESIGN 

cooling;  space  factor  (voltage);  and  also  on  the  t>T)e — 
whether  core  or  shell — since  this  affects  the  best  relation 
between  mean  lengths  of  the  copper  and  iron  circuits. 

The  Factor  kc.  Using  the  inch  for  the  unit  of  length, 
and  allowing  7  per  cent  for  eddy-current  losses  in  the 
copper,  the  resistivity  of  the  windings  will  be  0.9X10"^ 
ohms  per  inch-cube  at  a  temperature  of  80°  C;  the 
loss_  per    cubic    inch   of    copper  =  A2x 0.9X10"^,    and 

since  the  volume  is  2 1  —  jlc,  it  follows  that  kc  =  2X 

0.9  Xio-^ 

The  Factor  kt.  If  2^  =  total  watts  lost  per  pound  as 
read  off  one  of  the  curves  of  Fig.  27,  and  if  h  is  in  inches, 
we  have  the  equation 

whence 

0.2SW 


kt  = 


6.45/^" 


The  Factor  b.  The  ratio  of  full-load  copper  loss  to 
iron  loss  will  determine  the  load  at  which  maximum 
efl&ciency  occurs. 

Let  us  assume  the  k.v.a.  output  and  the  frequency  of 
a  given  transformer  to  be  constant,  and  determine  the 
conditions  under  which  the  total  losses  will  be  a  minimum. 
It  is  understood  that,  if  the  current  /  is  increased,  the 
voltage,  E,  must  be  decreased;  but  the  condition  k.v.a. 
=  EI  must  always  be  satisfied. 


PROCEDURE  IN  TRANSFORMER  DESIGN  145 

The  sum  of  the  losses  is  PFc+TTi;  but 

and 

Also,  since/  remains  constant,  EccB,  and' we  can  write 

TFi  =  a  constant  X£". 

The  quantity  which  must  be  a  minimum  is  therefore 

a  constant  ,  ^     <.  v^  m 
ha  constant X£. 

If  we  take  the  differential  coefficient  of  this  function 
of  E  and  put  it  equal  to  zero,  we  get  the  relation 

Wi       2 

The  value  of  n  for  high  densities  is  about  2,  while 
for  low  densities  it  is  nearer  to  1.7,  a  good  average 
being  1.85.  Thus,  to  obtain  maximum  efficiency  at 
full  load  in  a  power  transformer,   the  ratio  of  copper 

loss  to  iron  loss  should  be  about  &  =  -^  =0.925. 

2 

In  a  distributing  transformer,  in  order  to  obtain  a 
good  all-day  efficiency,  the  maximum  efficiency  should 
occur  at  about  f  full  load,  whence 

Wi        2 


146  PRINCIPLES  OF  TRANSFORMER  DESIGN 

Taking  w  =  1.75,  because  of  the  lower  densities  generally 
used  in  small  self-cooling  transformers,  we  get 

l        1-75X9  /  X 

b  =  — — — ^  =  1.97     or  (say)     2. 
4X2 

The  Ratio  j. .  Considerable  variations  in  this  ratio  are 

permissible,  even  in  transformers  of  a  given  type  wound 
for  a  particular  voltage,  and  that  is  one  reason  why  a 
close  estimate  of  the  volts  per  turn  as  given  by  Formula 
(49)  is  not  necessary.  Refmements  in  proportioning  the 
dimensions  of  a  transformer  are  rarely  justified  by  any 
appreciable  improvement  in  cost  or  efficiency;  a  certain 
minimum  quantity  of  material  is  required  in  order  to 
keep  the  losses  within  the  specified  limits;  but  consid- 
erable changes  in  the  shape  of  the  magnetic  and  electric 
circuits  can  be  made  without  greatly  altering  the  total 
cost  of  iron  and  copper,  provided  always  that  the  im- 
portant items  of  temperature  rise  and  regulation  are 
checked  and  maintained  within  the  specified  Umits. 

Figs.  49  and  50  show  the  assembled  iron  stampings  of 
single-phase  shell-  and  core-type  transformers.  The 
proportions  will  depend  somewhat  upon  the  voltage  and 
method  of  cooling;  but  if  the  leading  dimensions  are 
expressed  in  terms  of  the  width  (L)  of  the  stampings 
under  the  coils,  they  will  generally  be  within  the  following 
Umits: 

Shell  Type.  Core  Type. 

5  =  2  to  3  times  L  5  =  i  to  i .  8  times  L 

5  =  0.5  to  0.75  times  L  5  =  i  to  i  .5  times  L 

D  =  o.6  to  1.2  times  L  D  =  i  to  2  times  L 

H  =  i.2  to  3 . 5  times  L  Z^  =  3  to  6  times  L 


PROCEDURE  IN  TRANSFORMER  DESIGN 


147 


By  taking  the  averages  of  these  figures,  and  roughly- 
approximating  the  lengths  U  and  /;  in  each  case,  the  mean 
value  of  the  required  ratio  is  found  to  be 

y  =  1 . 2  (approx.)  for  shell  type,  | 
Y  =  o.s  (approx.)  for  core  type. 


Fig.  49. — Assembled  Stampings  of  Single-phase  Shell-type  Transformer. 

Having  determined  the  values  of  the  various  quan- 
tities appearing  in  Formula  (53),  it  is  now  possible 
to   calculate   an   approximate   average   value   for   the 

quantity  ^  and  for  the  coefficient  c  of  Formula  (49). 

We  shall  make  the  further  assumptions  (refer  Art.  20) 
that  A  =  1100  amperes  per  square  inch,  and  5  ^8000 


148 


PRINCIPLES  OF  TRANSFORMER  DESIGN 


gausses;  the  transformer  being  of  the  shell  t^-pe  for  use 
on  distributing  circuits  of  frequency  60.  Then,  by 
Formula  (53), 


/$_  2X0.9X1 100X1.2X6.45X60X9000 


TI 


io''X  2X0.28X0.75 


=  19,720 


^ 

^ 

// 

E 
1- 

\ 

/ 

H 

- 

Fig.  50. — Assembled  Stampings  of  Single-phase  Core-type  Transformer. 

wherein  the  figure  0.75  is  the  value  of  w  read  of!  the 
curve  for  sihcon  steel  in  Fig.  27. 

The  value  of  the  coefficient  in  Formula  (49),  for  the 
assumed  conditions,  is  therefore 


=  yP^  X  19,720  =  0.0296. 


PROCEDURE  IN  TRANSFORMER  DESIGN  149 

Similarly,  for  a  core-type  power  transformer;  if /=25. 
5  =  13,000,  and  A  =  1350,  we  have, 

/'l>^  2X0.9X1350X0.5X6.45X25X13,000^ 

TI  106X0.925X0.28X0.58  '^^  ' 

Whence  c  =  0.02  74. 

Having  shown  what  factors  determine  this  design 
coefficient,  it  will  merely  be  necessary  to  give  a  hst  of 
values  from  which  a  selection  should  be  made  for  the 
purpose  of  calculating  the  quantity  Vt  of  Formula  (49a). 

For  shell-type  power  transformers  c  =  o.o4  to  0.045 

For  shell-type  distributing  transformers  ^  =  0.03 
For  core-type  power  transformers  c  =  0.025  to  0.03 

For  core-type  distributing  transformers  c  =  o.o2 

Where  a  choice  of  two  values  of  c  is  given,  the  lower 
value  should  be  chosen  for  transformers  wound  for  high 
pressures.  When  the  voltage  is  low  the  value  of  c  is 
shghtly  higher  because  of  the  alteration  in  the  ratio. 

Ic 

-  which  depends  somewhat  on  the  copper  space  factor. 
h 

The  proposed  values  here  given  for  this  design  coeflS- 
cient  are  based  on  the  assumption  that  silicon  steel 
stampings  are  used  in  the  core.  If  ordinary  trans- 
former iron  is  used — as,  for  instance,  in  small  distrib- 
uting transformers — it  will  be  advisable  to  take  about  f 
of  the  above  values  for  the  coefficient  c. 

40.  Procedure  to  Determine  Dimensions  of  a  New 
Design.  With  the  aid  of  the  design  coefficient  c,  it  is 
now  possible  to  calculate  the  number  of  volts  that  should 


150  PRINCIPLES  OF  TRANSFORMER  DE$IGN 

be  generated  in  one  turn  jf  the  winding  of  a  transformer 
of  good  design  according  to  present  knowledge  and  prac- 
tice. The  logical  sequence  of  the  succeeding  steps  in 
the  design,  may  be  outlined  as  follows: 

(i)  Determine  approximate  dimensions. 

(a)  Calculate  volts  per  turn  by  Formula  (49). 

(6)  Assume  current  density  (select  suitable  trial 
value  from  table  in  Art.  20).  Decide  on 
number  of  coils.  Calculate  cross-section 
of  copper. 

(c)  Decide  upon  necessary  insulation  and  oil- 

or  air-ducts  between  coils,  and  between 
windings  and  core.  Determine  shape  and 
size  of  "  window  "  or  opening  necessary  to 
accommodate  the  windings. 

(d)  Calculate  total  flux  required.     Assume  flux 

density  (select  suitable  trial  value  from 
table  in  Art.  20),  and  calculate  cross-sec- 
tion of  core.  Decide  upon  shape  and  size 
of  section,  including  oil-  or  air-ducts  if 
necessary. 

(e)  Calculate  iron  and  copper  losses,  and  modify 

the  design  slightly  if  necessary  to  keep 
these  within  the  specified  limits. 

(2)  Calculate  approximate  weight  and  cost  of  iron 
and  copper  if  desired  to  check  with  permissible  maximum 
before  proceeding  with  the  design. 

(3)  Calculate  exciting  current. 

(4)  Calculate  leakage  reactance  and  voltage  regula- 
tion. 


PROCEDURE  IN  TRANSFORMER  DESIGN  151 

(5)  Calculate  necessary  cooling  surfaces.  Design  con- 
taining tank  and  lid,  providing  not  only  sufficient  oil 
capacity  and  cooling  surface,  but  also  the  necessary 
clearances  to  insure  proper  insulation  between  current- 
carrying  parts  and  the  case.     Calculate  temperature  rise. 

41.  Space  Factors.  The  copper  space  factor,  as  pre- 
viously defined  (see  Art.  15),  is  the  ratio  between  the 
cross-section  of  copper  and  the  area  of  the  opening  or 
"  window "  which  is  necessary  to  accommodate  this 
copper  together  with  the  insulation  and  oil-  or  air-ducts. 
It  may  vary  between  0.55  in  transformers  for  use  on 
circuits  not  exceeding  660  volts,  to  0.06  in  power  trans- 
formers wound  for  about  100,000  volts.  An  estimated 
value  of  the  probable  copper  space  factor  may  be  useful 
to  the  designer  when  deciding  upon  one  of  the  dimensions 
of  the  "  window  "  in  the  iron  core.  For  this  purpose, 
the  curves  of  Fig.  51  may  be  used,  although  the  best 
design  and  arrangement  of  coils  and  ducts  will  not  always 
lead  to  a  space  factor  falling  within  the  limits  included 
between  these  two  curves. 

Iron  Space  Factor.  The  so-called  stacking  factor  for 
the  iron  core  will  be  between  0.86  and  0.9,  and  the  total 
thickness  of  core,  multiplied  by  this  factor,  will  give 
the  net  thickness  of  iron  if  there  are  no  oil-  or  air-ducts. 
When  spaces  are  left  between  sections  of  the  core  for 
air  or  oil  circulation,  the  iron  space  factor  may  be  from 
0.65  to  0.75. 

42.  Weight  and  Cost  of  Transformers.  The  weight 
per  k.v.a.  of  transformer  output  depends  not  only  upon 
the  total  output,  but  also  upon  the  voltage  and  fre- 
quency.    The  net  and  gross  weights  of  particular  trans- 


152 


PRINCIPLES  OF  TRANSFORMER  DESIGN 


1/ 

S 

1/ 

1 

^ 

1. 

11 

1, 

I 

1 

1 

J 

1 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

y^ 

/ 

/ 

X 

1 

c 

c 

\ 

5 

S5 

5 

\ 

<: 

\ 

5 

§1 


joioBj  aoBda  jaddoo 


PROCEDURE  IN  TRANSFORMER  DESIGN  153 

formers  can  be  obtained  from  manufacturers'  catalogues 
and  also  from  the  Handbooks  for  Electrical  Engineers. 
The  effect  of  output  and  frequency  on  the  weight  of  a 
line  of  transformers  designed  for  a  particular  voltage 
(in  this  instance,  22,000  volts)  is  roughly  indicated  by 
the  following  figures  of  weight  per  k.v.a.  of  output. 
These  figures  include  the  weight  of  oil  and  case. 

_,  ,     f  100  k.v.a.  output    40  lb. 

Frequency  60  /  ^  ,u 

[  500  k.v.a.  output     23  lb. 

^  f  100  k.v.a.  output     S2  lb. 

Frequency  25  -^         ,  ^        ^    „ 

[  500  k.v.a.  output    35  lb. 

The  cost  of  transformers,  depending  as  it  does  on  the 
fluctuating  prices  of  copper  and  iron,  is  very  unstable. 
Within  the  last  few  years,  the  variation  in  the  price  of 
copper  wire  has  been  about  100  per  cent,  and  the  cost  of 
the  laminated  iron  for  the  cores  has  also  undergone  great 
changes.  The  best  that  can  be  done  here  is  to  indicate 
how  the  cost  depends  upon  voltage  and  output.  That  a 
high  frequency  always  means  a  cheaper  transformer  is 
evident  from  an  inspection  of  the  fundamental  Formula 
(48)  of  Art.  37.  If  /  is  increased,  either  $,  or  {TI),  or 
both,  can  be  reduced,  and  this  means  a  saving  of  iron, 
or  copper,  or  both.  The  effect  of  an  increase  in  voltage 
is  felt  particularly  in  the  smaller  sizes,  but  an  increase 
of  voltage  always  means  an  addition  to  the  cost;  while 
an  increase  of  size  for  a  given  voltage  results  in  a  reduc- 
tion of  the  cost  per  k.v.a.  of  output. 

Some  idea  of  the  dependence  of  cost  on  output  and 
voltage  may  be  gained  from  the  fact  that  the  unit  cost 


154  PRINCIPLES  OF  TRANSFORMER  DESIGN 

would  be  about  the  same  for  (i)  a  1500  k.v.a.  trans- 
former wound  for  22,000  volts,  (2)  a  2000  k.v.a.  trans- 
former wound  for  44,000  volts,  and  (3)  a  3000  k.v.a, 
transformer  wound  for  88,000  volts. 

Three-phase  Transformers.  It  does  not  appear  to  be 
necessary  to  supplement  what  has  been  said  in  Articles 
5  and  8  on  the  subject  of  three-phase  transformers. 
Once  the  principles  underlying  the  design  of  single-phase 
transformers  are  thoroughly  understood,  it  is  merely 
necessary  to  divide  any  polyphase  transformer  (see 
Figs.  12,  13,  and  14)  into  sections  which  can  be  treated 
as  single-phase  transformers,  due  attention  being  paid 
to  the  voltage  and  k.v.a.  capacity  of  each  such  unit 
section  of  the  three-phase  transformer. 

The  saving  of  materials  effected  by  combining  the 
magnetic  circuits  of  three  single-phase  transformers 
so  as  to  produce  one  three-phase  unit,  usually  results 
in  a  reduction  of  10  per  cent  in  the  weight  and 
cost. 

43.  Numerical  Example.  It  is  proposed  to  design  a 
single-phase  1500  k.v.a.  oil-insulated,  water-cooled, 
transformer  for  use  on  an  88, 000- volt  power  transmission 
system.  A  design  sheet  containing  more  detailed  items 
than  would  generally  be  considered  necessary  will  be 
used  in  order  to  illustrate  the  various  steps  in  the  design 
as  developed  and  discussed  in  the  preceding  articles. 
Two  columns  will  be  provided  for  recording  the  known  or 
calculated  quantities,  the  first  being  used  for  preliminary 
assumptions  or  tentative  values,  while  the  second  will  be 
used  for  final  results  after  the  preliminary  values  have 
been  either  confirmed  or  modified. 


PROCEDURE  IN  TRANSFORMER  DESIGN  155 

Specification 

Output 1,500  k.v.a. 

Number  of  phases one 

H.T.  voltage 88,000 

L.T.  voltage 6,600 

Frequency 5° 

Maximum  efficiency,  to  occur  at  full  load 

and  not  to  be  less  than 98-1% 

Voltage  regulation,  on  80  per  cent  power 
factor 5% 

Temperature  rise  after  continuous  full- 
load  run 40    C. 

Test  voltage:   H.T.  winding  to  case  and 

L.T.  coils 177,000 

L.T.  winding  to  case 14,000 

The  calculated  values  of  the  various  items  are  here 
brought  together  for  reference  and  for  convenience  in 
following  the  successive  steps  in  the  design.  The  items 
are  numbered  to  facilitate  reference  to  the  notes  and  more 
detailed  calculations  which  follow. 

Items  (i)  and  (2).— L.T.  Winding.  By  Formula  (49a), 
Art.  39,  page  142,  the  volts  per  turn,  for  a  shell-type 
power  transformer,  are 

F,  =  o.o42Vi,5oo,ooo  =  5i.5, 
whence, 

51-5 


156 


PRIN'CIPLES  OF  TRANSFORMER  DESIGN 


DESIGN  SHEET 


[.  Volts  per  turn. 


L.T.  Winding  (Secondary) 

Total  number  of  turns 

Number  of  coils 

Number  of  turns  per  coil 

Secondary  current,  amperes 

Current  density,  amperes  per  sq.  in. .  . 
Cross-section  of  each  conductor,  sq.  in. 

8.  Insulation  on  wire,  cotton  tape,  in.  .  . 

9.  Insulation  between  layers,  in 

10.  Number  of  turns  per  layer,  per  coil. .  . 

11.  Number  of  layers 

12.  Overall  width  of  finished  coil  (say),  in. 

13.  Thickness    (or   depth)    of   coil,    with 

allowance     for     irregularities     and 
bulging  at  center,  in 


H.T.  Winding  (Primary) 

Total  number  of  turns 

Number  of  coils 

Number  of  turns  per  coil 

Primary  current,  amperes 

Current  density,  amperes  per  sq.  in . 
Cross-section  of  each  wire,  sq.  in .  .  . 
Insul.  on  wire  (cotton  covering) ,  in . . 


Symbol. 


Assumed 
or  Approxi- 
mate 
Values. 


128 

6 
21.3 


Final 
Values. 


52.3 


126 
6 


227 
1575 


1600 
3strips,  eacho.i6Xo.3=o.r44 

j    0.026 

(2X0. 006) +0.01 2  =  0.024 


21 
0.36 


"5 


1680 
18 


o  in  2  coils;  95  in  16  coils 
Ip      I     17-05     j 
A        I       1640      I 
0.04X0.26  =  0.0104 
2X0.008  =  0.016 


21.  Insul.  between  layers,  fullerboard,  in. . 

0.012 

22.  Number  of  turns  per  layer,  per  coil..  . 

I 

23.  Number  of  layers;  in  all  but  end  coils 

95 

24.  Overall  width  of  finished  coil,  in 

0.31 

25.  Thickness  or  depth  of  coil,  in 

6-75 

26.  Make  sketch  of  assembly  of  coils,  with 

necessary  insulating  spaces  and  oil 

ducts. 

PROCEDURE  IN  TRANSFORMER  DESIGN 


157 


DESIGN  SHEET— Continued 


Symbol. 


Assumed 
or  Approxi- 
mate 
Values. 


Final 
Values. 


27.  Size   of   "  window "   or  opening   for 
windings,  in 


Magnetic  Circuit 

28.  Total  flux  (maxwells) 

29.  Maximum   value   of   flux   density   in 

core  under  windings  (gausses) 

30.  Cross-section  of  iron  under  coils,  sq.  in. 

31.  Number  of  oil  ducts  in  core 

32.  Width  of  oil  ducts  in  core 

33.  Width  of  stampings  under  windings, 


34.  Net  length  of  iron  in  core,  in . 
36. 


37. 


38. 


39- 


Gross  length  of  core,  in 

Cross-section  of  iron  in  magnetic  cir- 
cuit outside  windings,  sq.  in 

Flux  density  in  core  outside  windings 
(gausses) 

Average  length  of  magnetic  circuit 
under  coils,  in 

Average  length  of  magnetic  circuit 
outside  coils,  in 

Weight  of  core,  lb 

Losses  in  the  iron,  watts 


Copper  Losses 

42.  Mean  length  per  turn  of  primary,  ft. . 

43.  Resistance  of  primary  winding,  ohms . 

44.  Full-load  losses  in  primary  (exc.  cur- 

rent neglected) 

45.  Mean  length  per  turn  of  secondary,  ft. 

46.  Resistance  of  secondary  winding,  ohms 

47.  Full-load  losses  in  secondary,  watts  .  . 

48.  Total  full-load  copper  losses,  watts . . 


12.75X32 


2.36X10' 

13,000 
282 
none 


Wi 


Wc 


13,850 
264 


II     ^ 
24 

27. 

264 

13,850 

32. 

79- S 
8250 
11,900 


10.15 
18. 1 

5360 
10.15 
o . 0962 

4960 
10,220 


158 


PRINCIPLES  OF  TRANSFORMER  DESIGN 


DESIGN  SHEET— Continued 


49.  Total  weight  of  copper  in  windings,  lb 

50.  Eflkiency  at  full  load  (unity  power 

factor) 

51.  Efficiency  at  other  loads  and  power 

factors  (refer  to  text  following). 

52.  No-load    primary    exciting    current 

amperes 


Regulation 
53.  Reactive  voltage  drop.  . 


54.  Equivalent  ohmic  voltage  drop .  .  . 

55.  Regulation  on  unity  power  factor,  per 

cent 

56.  Regulation  on  80  per  cent  power  fac 

tor,  per  cent 


Design  of  Tank,  Cooling  Surfaces 

57.  Effective  cooling  surface  of  tank,  sq.  in. 

58.  Number  of  watts  dissipated  from  tank 

surface 

59.  Watts  to  be  carried  away  by  circula- 

ting water 

60.  Size  and  length  of  pipe  in  cooling  coil,. 

61.  Approximate  flow  of  water  per  minute, 

gallons 

62.  Approximate  weight  of  oil,  lb 

63.  Estimated  total  weight  of  transformer, 

lb 


Symbol. 


Assumed 
or  Approxi- 
mate 
Values. 


7iRp 


19,360 

4650 

17,470 
is"X37o' 


■37 


Final 
Values. 


1700 
0.985 

2  IS 

2850 
600 

0-73S 
2.5 

18,860 


7300 


Items  (3)  and  (4).  The  number  of  separate  coils  is 
determined  by  the  following  considerations: 

(a)  The  voltage  per  coil  should  preferably  not  exceed 
5000  volts. 


PROCEDURE  IN  TRANSFORMER  DESIGN  159 

(b)  The  thickness  per  coil  should  be  small  (usually 
within  1.5  in.)  in  order  that  the  heat  may  readily  be 
carried  away  by  the  oil  or  air  in  the  ducts  between  coils 
(Refer  Art.  23). 

(c)  The  number  of  coils  must  be  large  enough  to  admit 
of  proper  subdivision  into  sections  of  adjacent  primary 
and  secondary  coils  to  satisfy  the  requirements  of  regu- 
lation by  Hmiting  the  magnetic  flux-linkages  of  the  leak- 
age field. 

(d)  An  even  number  of  L.T.  coils  is  desirable  in  order 
to  provide  for  a  low-tension  coil  near  the  iron  at  each  end 
of  the  stack. 

To  satisfy  (a),  *here  must  be  at  least  — '- —  or,  sav- 

5000  ^ 

18  H.T.  coils.  If  an  equal  number  of  secondary  coils 
were  provided,  we  could,  if  desired,  have  as  many  as 
eighteen  similar  high-low  sections  which  would  be  more 
than  necessary  to  satisfy  (c).  The  number  of  these 
high-low  sections  or  groupings  must  be  estimated  now 
in  order  that  the  arrangement  of  the  coils,  and  the  num- 
ber of  secondary  coils,  may  be  decided  upon  with  a  view 
to  calculating  the  size  of  the  "  windows  "  in  the  mag- 
netic circuit.  It  is  true  that  the  calculations  of  reactive 
drop  and  regulation  can  only  be  made  later;  but  these 
will  check  the  correctness  of  the  assumptions  now  made, 
and  the  coil  grouping  will  have  to  be  changed  if  neces- 
sary after  the  preliminary  design  has  been  carried  some- 
what farther.  The  least  space  occupied  by  the  insula- 
tion, and  the  shortest  magnetic  circuit,  would  be  obtained 
by  grouping  all  the  primary  coils  in  the  center,  with  half 
the  secondary  winding  at  each  end,  thus  giving  only 


160  PRINCIPLES  OF  TRANSFORMER  DESIGN 

two  high-low  sections;  but  this  would  lead  to  a  very  high 
leakage  reactance,  and  regulation  much  worse  than  the 
specified  6  per  cent.  Experience  suggests  that  about  six 
high-low  sections  should  suffice  in  a  transformer  of  this 
size  and  voltage,  and  we  shall  try  this  by  arranging  the 
high-tension  coils  in  groups  of  six,  and  providing  sbc 
secondary  coils  (see  Fig.  52).  This  gives  us  for  item 
(4),  -^^  =  21.3  or,  say,  21,  whence  rs=i26. 
Items   (5)   to   (13).     The    secondary  current   is  Is  = 

-^ — '- =  227    amperes.     From    Art.     20,    we    select 

6600 

A  =  1600  as  a  reasonable  value  for  the  current  density, 

giving  --^=0.142  sq.  in.  for  the  cross-section  of   the 
1600 

secondary  conductor. 

In  order  to  decide  upon  a  suitable  width  of  copper 
in  the  secondary  coils,  it  will  be  desirable  to  esti- 
mate the  total  space  required  for  the  windings  so 
that  the  proportions  of  the  "  window  "  may  be  such 
as  have  been  found  satisfactory  in  practice.  The 
space  factor  (Art.  41)  is  not  likely  to  be  better  than 
0.1,    which    gives    for    the    area    of    the    "  window " 

—^  =  358  sq.  in.  Also,  if  a  reasonable  as- 
sumption is  that  H  =  2.$  times  D  (see  Fig.  49,  page  147), 
it  follows  that  2.5Z)XZ)  =  358;  whence  Z)  =  i2  inches. 

The  clearance  between  copper  and  iron  under  oil, 
for  a  working  pressure  of  6600  volts  (Formula  (12), 
Art.  16),  should  be  about  0.25+0.05X6.6  =  0.58  in. 
For  the  insulation  between  layers,  we  might  have 
0.02  in.  for  cotton,  and  a  strip  of  0.012  in.  fullerboard, 


PROCEDURE  IN  TRANSFORMER  DESIGN  161 

making  a  total  of  21X0.032  =  0.67  in.  The  thickness 
of  each  secondary  conductor  will  therefore  be  about 

12  — (0.58  +  0.67+0.58)  o       .  1..   u       •  -j^i- 

^-^^ ^-^=0.485  in.,  which  gives  a  width 

21 

of        ^  =0.293   in.     Let  us  make   this  0.3   in.,   and 
0.485 

build  up  each  conductor  of  three  strips  0.16  in.  thick, 
with  0.006  paper  between  wires  (to  reduce  eddy  cur- 
rent loss)  and  cotton  tape  outside.  Allowing  0.026  in. 
for  the  cotton  tape,  and  0.012  in.  for  a  strip  of  fuller- 
board  between  turns,  the  total  thickness  of  insulation, 
measured  across  the  layers,  is  21 X  (0.026+0.024)  = 
1.05  in. 

A  width  of  "  window "  of  12.75  ^^-  (see  Fig.  52) 
will  accommodate  these  coils.  The  current  density 
with  this  size  of  copper  is 

^  =  J3<5^^  =  '"5amps.persq.in. 

Items   (14)   to   (25).    H.T.   Winding.     Tp  =  i26X  — 

66 

=  1680.     This  may  be  divided  into  16  coils  of  95  turns 

each,  and  2  coils  of  only  80  turns  each,  which  would 

be  placed  at  the  ends  of  the  winding  and  provided  with 

extra  insulation  between  the  end  turns  (see  Art.  14), 

According  to  Formula  (13)  of  Ait.  16,  the  thickness 

of   insulation — consisting   of   partitions   of    fullerbcard 

with  spaces  between  for  oil  circulation — separating  the 

H.T.  copper  from  L.T.  coils  or  grounded  ircn,  should 

not  be  less  than  0.25+0.03X88  =  2  89  in.     Let  us  make 

this  clearance  3  in.     Then,  since  the  width  of  opening 


162  PRINCIPLES  OF  TRANSFORMER  DESIGN 

is  12.75  ^^•'  the  maximum  permissible  depth  of  winding 
of  the  primary  coils  will  be   12.75  —  6  =  6.75  in.     The 

primary  current  (Item  17)  is  Ip  =  —^ =  i7-05  amps. 

88,000 

(approx.).     The    cross-section    of    each    wire    is        '  ^ 

1600 

=  0.01065  sq.  in.     Allowing  0.016  in.  for  the  total  increase 

of  thickness  due  to  the  cotton  insulation,   and  0.012 


0^ 

<— 3'^  <-l,7'> 

g 
5 

Ire 

\x\x\ 

1  ■ 



II  1^  II 

1 

/      ^ 

Mill  III 

1 

m 

T 

11/ 

< 16-^^ > 

a 

Fig.  52. — Section  through  Windings  and  Insulation. 


in.  for  a  strip  of  fullerboard  between  turns,  the  thick- 
ness of  the  copper  strip  (assuming  flat  strip  to  be  used) 


must    not    exceed 


©- 


028  =  0.043     in.,     which 


makes  the  width  of  copper  strip  equal  to  -^ ^  =  o.  248 

0.043 

in.     Try  copper  strip  0.26X0.04  =  0.0104  sq.  in.,  makmg 

A  =  1640. 


PROCEDURE  IN  TRANSFORMER  DESIGN  163 

The  two  end  coils,  with  fewer  turns,  would  be  built 
up  to  about  the  same  depth  as  the  other  coils  by  putting 
increasing  thicknesses  of  insulation  between  the  end 
turns.  Thus,  since  there  is  a  total  thickness  of  copper 
equal  to  0.04  X  (95  —  80)  =0.6  in.  to  be  replaced  by  insula- 
tion, we  might  gradually  increase  the  thickness  of  fuller- 
board  between  the  last  eight  turns  from  0.012  in.  to 
0.15  in. 

Items  (26)  and  (27).  Size  of  Opening  for  Windings. 
A  drawing  to  a  fairly  large  scale,  showing  the  cross- 
section  through  the  coils  and  insulation,  should  now  be 
made.  Oil  ducts  not  less  than  j  in.  or  ys  in.  wide 
should  be  provided  near  the  coils  to  carry  off  the  heat, 
and  the  large  oil  spaces  between  the  H.T.  coils  and  the 
L.T.  coils  and  iron  stampings,  should  be  broken  up  by 
partitions  of  pressboard  or  other  similar  insulating 
material,  as  indicated  roughly  in  a  portion  of  the  sketch, 
Fig.  52.  In  this  manner  the  second  dimension  of  the 
"  window  "  is  obtained.  This  is  found  to  be  32  in., 
whence  the  copper  space  factor  is 

(i68oXo.oi04)  +  (i26Xo.i445)^^^g 
12.75X32 

Items  (28)  to  (41).  The  Magnetic  Circuit.  By 
Formula  (i).  Art.  2, 

88,000X108  .^^     _ 

^  = 77—  =  2.36  X 10^. 

4.44X50X1680       ^ 

Before  assuming  a  flux  density  for  the  core,  let  us 
calculate  the  permissible  losses. 


164  PRINCIPLES  OF  TRANSFORMER  DESIGN 

The  full  load  efficiency  being  0.981,  the  total  losses 

1, 500,000  X(i— 0.981)  ^^        ., 

are   —^ — ^  =  29,000   watts.     Also,   since 

0.901 

the  ratio   f— ^j   is  approximately  0.925   (see  Art.  39, 

under  sub-heading  The  Factor  b),  it  follows  that 

,„     29,000  ,, 

W,  = =  1 5 ,  100  watts, 

1.925 

whence 

I'Fc  =  29,000—  15,100  =  13,900  watts. 

Let  us  assume  the  width  of  core  under  the  windings 
(the  dimension  L  of  Fig.  49)  to  be  11  in.  and  the  width, 
B,  of  the  return  circuit  carrying  half  the  flux,  to  be 
5.5  in.  Then  the  average  length  of  the  magnetic  circuit, 
measured  along  the  flux  Hnes,  will  be  2(12.75  +  5.5+32  + 
5.5)  =  111.5  in. 

If  the  flux  density  is  taken  at  13,000  gausses  (selected 
from   the   approximate  values  of   Art.    20)    the  cross- 

o    "2  0  \^  T  Cy 

section  of  the  iron  is  — — —  =  282  sq.   in.     The 

13,000X0.45 

watts  lost  per  pound  (from  Fig.  27)  are  2^  =  1.27,  whence 

the  total  iron  loss  is 

PFi  =  1.27X0.28X282X111.5  =  11, 200  watts, 

which  is  considerably  less  than  the  permissible  loss.  It  is 
not  advisable  to  use  flux  densities  much  in  excess  of  the 
selected  value  of  13,000  gausses  for  the  following  reasons: 
(a)  The  distortion  of  wave  shapes  when  the  mag- 
netization is  carried  beyond  the  "  knee  "  of  the  B-H 
curve. 


PROCEDURE  IN  TRANSFORMER  DESIGN  163 

(6)  The  large  value  of  the  exciting  current. 

(c)  The  difficulty  of  getting  rid  of  the  heat  from  the 
surface  of  the  iron  when  the  watts  lost  per  unit  volume 
are  considerable. 

Let  us,  therefore,  proceed  with  the  design  on  the  basis 
of  i4,oco  gausses  as  an  upper  limit  for  the  flux  density. 

If  no  oil  ducts  are  provided  between  sections  of  the 
stampings,  the  stacking  factor  will  be  about  0.89.  A 
gross  length  of  27  in.  (Item  35)  gives  24  in.  for  the  net 
length,  and  a  cioss-section  of  24  X 1 1  =  264  sq.  in.  Whence 
.6  =  13,850  gausses,  and  the  total  weight  of  iron  is 
264X111.5X0.28  =  8250  lb. 

The  watts  per  pound,  from  Fig.  27,  are  ^  =  1.44, 
whence  P^«  =  11,900. 

Items  (42)  to  (49),  Copper  Loss.  The  mean  length  per 
turn  of  the  windings  is  best  obtained  by  making  a  draw- 
ing such  as  Fig.  53.  This  sketch  shows  a  section  through 
the  stampings  parallel  with  the  plane  of  the  coils.  The 
mean  length  per  turn  of  the  secondary,  as  measured 
off  the  drawing,  is  122  in.,  and  since  the  length  per  turn 
of  the  primary  coils  will  be  about  the  same,  this  dimen- 
sion will  be  used  in  both  cases.  Taking  the  resistivity 
of  the  copper  at  0.9X10"^  ohms  per  inch  cube  (see  The 
Factor  kc,  in  Art.  39),  the  primary  resistance  (hot)  is 


J,      0.9  X 122X1680      -       , 

Ai=        f,,, =  18.1  ohms, 

10^X0.0104 


whence  the  losses  (Item  44)  are 

(17.05)2x18.1  =  5260  watts. 


166  PRINCIPLES  OF  TRANSFORMER  DESIGN 

For  the  secondary  winding  we  have 

„      0.QX122X126  .      , 

R2  =  ~^ =0.0062  ohm, 

io«Xo.i44 


if! 

/ 


-4'G- 


-I2?i- 


FiG.  53.— Section  through  Coil  and  Stampings. 

whence  the  losses  (Item  47)  are  (227)2x0.0962=4960 
watts,  and 

1^0  =  5360+4960  =10,220  watts, 

which  is  appreciably  less  than  the  permissible  copper  loss. 


PROCEDURE  IN  TRANSFORMER  DESIGN  167 

It  is  at  this  stage  of  the  calculations  that  changes 
should  be  made,  if  desirable,  to  reduce  the  cost  of  mate- 
rials, by  making  such  modifications  as  would  bring  the 
losses  near  to  the  permissible  upper  limit.  The  obvious 
thing  to  do  in  this  case  would  consist  in  increasing  the 
current  density  in  the  windings,  and  perhaps  making  a 
small  reduction  in  the  number  of  turns.  A  considerable 
saving  of  copper  would  thus  be  effected  without  neces- 
sarily involving  any  appreciable  increase  in  the  weight 
of  the  iion  stampings.  Since  this  example  is  being 
worked  through  merely  for  the  purpose  of  illustrating 
the  manner  in  which  fundamental  principles  of  design 
may  be  appHed  in  practice,  no  changes  will  be  made 
here  to  the  dimensions  and  quantities  already  calculated. 

The  weight  of  copper  (Item  49)  is 

0.32  (i22Xi68oXo.oio4)-(- 

0.32(122X126X0.144)  =  1,700  lb. 

Items  (50)  and  (51).  Efficiency.  The  full-load  effi- 
ciency on  unity  power  factor  is 


1 ,  1^00,000  o 

'^  -0.985. 


1 , 500,000 -h  1 1 ,900 -f- 1  o,  2  20 

The  calculated  efficiencies  at  other  loads  are: 

At  il  full  load 0.985 

At    f  full  load o. 984 

At    I  full  load 0.981 

At    i  full  load 0.968 


168  PRINCIPLES  OF  TRANSFORMER  DESIGN 

The  full-load  efficiency  on  80  per  cent  power  factor  is 

1,500,000X0.8  o 

'■^     '  =0.982. 


(1,500,000X0.8) +  22,120 


Item  (52).  Open-circuit  Exciting  Current.  Using 
the  curves  of  Fig.  45  (see  Art.  35  for  explanation),  we 
obtain  for  a  density  B  =  13,850  the  value  23  volt-amperes 
per  pound  of  core.  The  weight  of  iron  (Item  40)  being 
8250  lb.,  it  follows  that  the  exciting  current  is 

J.     8250X23 
88,000 

This  is  12.6  per  cent  of  the  load  component,  which  is 
rather  more  than  it  should  be.  If  the  design  is  altered, 
as  previously  suggested,  to  reduce  the  amount  of  copper, 
this  will  result  in  a  reduction  of  the  opening  in  the  iron, 
and,  therefore,  also  of  the  length  of  the  magnetic  circuit. 
It  is,  however,  clear  that  the  flux  density  (Item  29)  must 
not  be  higher  than  13,850  gausses.  If  the  design  were 
modified,  it  might  be  advisable  to  reduce  this  value  by 
sKghtly  increasing  the  cross-section  of  the  magnetic 
circuit.  The  fact  that  the  exciting  current  component 
is  fairly  large  relatively  to  the  load  current  will  lead  to  a 
small  increase  in  the  calculated  copper  loss  (Item  44); 
but  for  practical  purposes  it  is  unnecessary  to  make  the 
correction. 

Items  (53)  to  (56)  Regulation.  Referring  to  Fig.  52,  it 
is  seen  that  there  are  six  high-low  sections,  all  about 
equal,  since  the  smaller  number  of  turns  in  two  out  of 
eighteen  primary  coils  is  not  worth  considering  in  calcu- 


PROCEDURE  IN  TRANSFORMER  DESIGN  169 

lations  which  cannot  in  any  case  be  expected  to  yield 
very  accurate  results.  The  quantities  for  use  in  Formula 
(40)  of  Art.  34  have,  therefore,  the  following  values: 

ri=^¥^  =  28o; 

/i  =  i7-os; 

/  =  io.i5Xi2X2. 54  =  310  cm.; 
5  =  3X2.54  =  7.62  cm.; 
/>  =  i. 7X2. 54  =  4.32  cm.; 
5  =  0.38X2.54  =  0.965  cm.; 
/f=i2. 75X2.54  =  32.4  cm. 

whence  the  induced  volts  per  section  are, 

/iZi=475  volts. 

Since  there  are  six  sections,  and  all  the  turns  are  in  series, 
the  total  reactive  drop  at  full  load  is 

/iXj, =475X6  =  2850  volts, 

which  is  only  3.24  per  cent  of  the  primary  impressed 
voltage. 

By  Formula   (35)   Art.  s^,  the  equivalent  primary 
resistance  is 

i?p  =  i8.i  +  ( — — j  X0.0962  =35.2  ohms; 

whence 

IiRp  =  600  volts. 

which  is  0.683  per  cent  of  the  primary  impressed  voltage. 
By  Formula  (47),  Art.  36,  when  the  power  factor  is 
unity  (cos  ^  =  1). 

Regulation  =  0.683 +0  =  0.683  P^^^  cent 


170  PRINCIPLES  OF  TRANSFORMER  DESIGN 

The  more  correct  value,  as  obtained  from  Formula  (46) 
is  0.735. 

When  the  power  factor  of  the  load  is  80  per  cent,  the 
approximate  formula — which  is  quite  sufficiently  accu- 
rate in  this  case — gives 

Regulation  =  (0.683  X0.8)  +  (3.24  X0.6) 

=  2.5  per  cent  (approx.)  on  80  per  cent  power  factor. 

This  is  very  low,  and  considerably  less  than  the  specified 
limit  of  5  per  cent.  It  is  possible  that  the  specified  reg- 
ulation might  be  obtained  with  only  4,  instead  of  6,  high- 
low  groups  of  coils,  and  in  order  to  produce  the  cheapest 
transformer  to  satisfy  the  specification,  the  designer 
would  have  to  abandon  this  preliminary  design  until  he 
had  satisfied  himself  whether  or  not  an  alternative 
design  with  a  different  grouping  of  coils  would  fulfill  the 
requirements.  It  is  clear  from  the  inspection  of  Fig.  52 
that  an  arrangement  with  only  four  L.T.  coils  and  (say) 
sixteen  H.T.  coils  would  considerably  reduce  the  size  of 
the  opening  in  the  stampings,  thus  saving  materials  and, 
incidentally,  reducing  the  magnetizing  current,  which  is 
abnormally  high  in  this  preliminary  design. 

Items  (57)  to  (61).  Requirements  for  Limiting  Tem- 
perature Rise.  A  plan  view  of  the  assembled  stampings 
should  be  drawn,  as  in  Fig.  54,  from  which  the  size 
of  containing  tank  may  be  obtained.  In  this  instance 
it  is  seen  that  a  tank  of  circular  section  5  ft.  3  in.  diam- 
eter will  accommodate  the  transformer.  The  heiglit 
of  the  tank  (see  Fig.  55)  will  now  have  to  be  estimated 
in  order  to  calculate  the  approximate  cooling  surface. 
This  height  will  be  about  90  in.,  and  if  we  assume  a 


PROCEDURE  IN  TRANSFORMER  DESIGN  171 

smooth  surface  (no  corrugations),  the  watts  that  can 
be  dissipated  continuously  are 

o.24x|^(7rX63X9o)+'^^J  =4650; 


Fig.  54. — Assembled  Stampings  in  Tank  of  Circular  Section. 


the   multiplier   0.24  being   obtained   from   the   curve, 
Fig.  32  of  Art.  25. 

The  watts  to  be  carried  away  by  the  circulating 
water  are  (10,220+11,900)— 4650  =  17,470.  From  data 
given  in  Art.  29,  it  follows  that  a  coil  made  of  ij  in. 


172 


PRINCIPLES  OF  TRANSFORMER  DESIGN 


tube  should  have   a   length   of   '-^ =  37©   ft. 

i2Xi.25X7r 


!h.  T.  Terminal  as 
aetaUedln  Fig.  26 


Fig.  55- — Sketch  of  1500-k.v.a.,  88,000-volt  Transformer  in  Tank. 


Assuming   the   coil   to   have   an   average   diameter   of 
4  ft.  8  in.,  the  number  of  turns  required  will  be  about  25. 


PROCEDURE  IN  TRANSFORMER  DESIGN  173 

On  the  basis  of  I  gal.  of  water  per  kilowatt,  the 
required  rate  of  flow  for  an  average  temperature  dif- 
ference of  15°  C.  between  outgoing  and  ingoing  water 
is  0.25X17.47  =  4.37  gal.  per  minute.  This  amount 
may  have  to  be  increased  unless  the  pipes  are  kept 
clean  and  free  from  scale. 

The  completed  sketch,  Fig.  55,  indicates  that  a 
tank  87  in.  high  will  accommodate  the  transformer 
and  cooling  coils,  and  the  corrected  cooling  surface  for 
use  in  temperature  calculations  (see  Art.  25)  is  therefore 

5  =  (7rX63X87)+-(-X63^j  =  i8,86o  sq.  in. 


This  new  value  for  Item  57  has  been  put  in  the  last 
column  of  the  design  sheet;  but  the  items  immediately 
following,  which  are  dependent  upon  it,  have  not  been 
corrected  because  the  difference  is  of  no  practical  im- 
portance. 

Hottest  Spot  Temperature.  The  manner  in  which  the 
temperature  at  the  center  of  the  coils  may  be  calculated 
when  the  surface  temperature  is  known,  was  explained 
in  Art.  23.  It  is  unnecessary  to  make  the  calculation 
in  this  instance  because  the  coils  are  narrow  and  built 
up  of  flat  copper  strip.  There  will  be  no  local  "  hot 
spots  "  if  adequate  ducts  for  oil  circulation  are  provided 
around  the  coils. 

Items  (62)  and  (63).  Weight  of  Oil  and  of  Complete 
Transformer.  The  weight  of  an  average  quahty  of 
transformer  oil  is  53  lb.  per  cubic  foot,  from  which  the 
total  weight  of  oil  is  found  to  be  about  7300  lb.    The 


174  PRINCIPLES  OF  TRANSFORMER  DESIGN 

calculated  weights  of  copper  in  the  windings  (Item  49) 
and  iron  in  the  core  (Item  40)  are  1700  lb.  and  8250  lb., 
respectively.  The  sum  of  these  three  figures  is  17,250  lb. 
This,  together  with  an  estimated  total  of  4750  lb.  to  cover 
the  tank,  base  and  cover,  cooling  coil,  terminals,  solid 
insulation,  framework,  bolts,  and  sundries,  brings  the 
weight  of  the  finished  transformer  up  to  22,000  lb.  (in- 

22  000 

eluding  oil) ;  or  — ^ =  14.65  lb.  per  k.v.a.  of  rated  full- 
load  output. 

Several  details  of  construction  have  not  been  referred 
to.  It  is  possible,  for  instance,  that  tappings  should  be 
provided  for  adjustment  of  secondary  voltage  to  com- 
pensate for  loss  of  pressure  in  a  long  transmission  line. 
These  should  preferably  be  provided  in  a  portion  of  the 
winding  which  is  always  nearly  at  ground  potential. 
It  is  not  uncommon  to  provide  for  a  total  voltage  varia- 
tion of  10  per  cent  in  four  or  five  steps,  which  is  accom- 
plished by  cutting  in  or  out  a  corresponding  number  of 
turns,  either  on  the  primary  or  secondary  side,  which- 
ever may  be  the  most  convenient. 

Mechanical  Stresses  in  Coils.  The  manner  in  which  the 
projecting  ends  of  flat  coils  in  a  shell-type  transformer 
should  be  clamped  together  is  shown  in  Fig.  16  of  Art.  9. 
Let  us  calculate  the  approximate  pressure  tending  to 
force  the  projecting  portion  of  the  secondary  end  coils 
outward  when  a  dead  short-circuit  occurs  on  the  trans- 
former. The  force  in  pounds,  according  to  Formula  (4), 
is 

J-  vi  max  -Oam 


8,896,000 


PROCEDURE  IN  TRANSFORMER  DESIGN  175 

For  the  quantities  T  and  /,  we  have 

0  O 

and  /,  being  the  average  length  of  the  portion  of  a 
turn  projecting  beyond  the  stampings  at  one  end,  is 

;  =  £2di><I£_27  =  34  in.  or  86  cms. 

2 

The  value  of  the  quantities  /^ax  and  B^^  depends  on 
the  impedance  of  the  transformer.  With  normal  full- 
load  current,  the  impedance  drop  is 


/i;^^  =  V(285o)2+ (600)2  =  2910  volts, 

where  the  quantities  under  the  radical  are  the  items  53 
and  54  of  the  design  sheet.  In  order  to  choke  back  the  full 
impressed  voltage,  the  current  would  have  to  be  about 

— '- =  (say)  thirty  times  the  normal  full-load  value. 

291C 

Thus  the  current  value  for  use  in  Formula  (4),  on  the  sine 
wave  assumption,  will  be 

/max  =  3oX227xV2  =  965o  ampcrcs. 

The  density  of  the  leakage  flux  through  the  coil  is  less 
easily  calculated;  but,  since  the  reactive  voltage  was 
calculated  on  the  assumption  of  flux  lines  all  parallel  to 
the  plane  of  the  coil,  we  may  now  consider  a  path  one 
square  centimeter  in  cross- section  and  of  length  equal  to 


176  PRINCIPLES  OF  TRANSFORMER  DESIGN 

the  depth  of  the  coil  (about  29  cms.)  in  which  the  leakage 
flux  will  have  the  average  value. 

^8111  =  -  — X21X9650X—   =4400  gausses, 
2L10  29J 

whence,  by  Formula  (4), 

Force  in  lb.  =  "X86X965°X44°o ^ gg      ^^ 

8,896,000 

This  is  the  force  F  of  Fig.  16,  distributed  over  the  whole 
of  the  exposed  surface  of  the  end  coil.  An  equal  force 
will  tend  to  deflect  outward  the  secondary  coil  at  the 
other  end  of  the  stack.  If  an  arrangement  of  straps  with 
two  bolts  is  adopted — as  shown  in  Fig.  16 — each  bolt 
must  be  able  to  withstand  a  maximum  load  of  4370  lb. 
Bolts  I  in.  diameter  will,  therefore,  be  more  than  suf- 
ficient to  prevent  displacement  of  the  coils,  even  on  a 
dead  short  circuit. 


CHAPTER  VI 

TRANSFORMERS  FOR  SPECIAL  PURPOSES 

44.  General  Remarks.  When  applying  the  funda- 
mental principles  of  electrical  design  to  special  types  of 
apparatus,  it  is  necessary  to  consider  what  are  the  chief 
characteristics  of  such  apparatus  and  wherein  they  differ 
from  those  of  the  more  usual  types.  The  apparatus  dealt 
with  in  the  preceding  chapters  is  the  potential  transformer 
for  use  either,  as  large  units,  in  power  stations,  or  in 
smaller  sizes,  as  means  of  distributing  electric  power  in 
residential  or  industrial  districts.  A  few  special  types 
of  transformer  will  now  be  considered;  but  the  treat- 
ment will  be  brief,  with  the  object  of  avoiding  useless 
repetitions.  Attention  will  be  given  mainly  to  such  dis- 
tinctive features  or  peculiarities  as  may  have  an  impor- 
tant bearing  on  the  design, 

45.  Transformers  for  Large  Currents  and  Low  Volt- 
ages. Electric  furnaces  are  built  to  take  currents  up 
to  35,000  amperes  at  about  80  volts — usually  three-phase. 
Welding  transformers  must  give  large  currents  at  a  com- 
paratively low  voltage.  A  current  of  2000  amperes  at 
5  volts  would  probably  be  required  for  rail  welding  on 
an  electric  railroad.  Transformers  for  thawing  out 
frozen  water  pipes  need  not  necessarily  be  specially 
designed  because  standard  distributing  transformers — 

177 


178  PRINCIPLES  OF  TRANSFORMER  DESIGN 

connected  to  give  about  50  volts — are  used  successfully 
for  this  purpose.  A  transformer  of  12  k.v.a.  normal 
rating,  capable  of  giving  up  to  6cx>  amperes  with  a  max- 
imum pressure  of  30  volts  for  short  periods  of  time  in 
cold  weather,  will  probably  answer  all  requirement  for 
the  thawing  of  house  service  pipes  up  to  i^  in.  diameter. 
A  current  of  400  amperes  will  thaw  out  a  i-in.  pipe  in 
about  half  an  hour. 

In  the  design  of  all  transformers  for  large  currents, 
especially  when  they  are  Hable  to  be  practically  short- 
circuited,  the  leakage  reactance  (see  Art.  34)  is  a  matter 
of  importance.  The  permissible  maximum  current  on  a 
short  circuit  should  be  specified.  In  some  cases,  sepa- 
rate adjustable  reactance  coils  (usually  on  the  high- 
voltage  side)  are  provided  for  the  purpose  of  regulating 
the  current  from  transformers  used  for  welding  and  sim- 
ilar processes. 

Another  point  to  be  watched  in  the  design  of  trans- 
formers for  large  currents  is  the  eddy  current  loss  in  the 
copper  (see  Art.  20),  which  must  be  minimized  by  prop- 
erly arranging  and  laminating  the  secondary  winding 
and  leads.  The  mechanical  details  in  the  design  of 
secondary  terminals  and  leads  also  require  careful  atten- 
tion. 

46.  Constant-current  Transformers.  Circuits  with 
incandescent  or  arc  lamps  connected  in  series  require  the 
amount  of  current  to  be  approximately  constant  regard- 
less of  the  number  of  lamps  on  the  circuit.  If  it  is  de- 
sired to  supply  series  circuits  of  this  nature  from  constant 
potential  mains,  special  transformers  are  required,  so 
designed  as  to  give  variable  voltage  at  the  secondary 


TRANSFORMERS  FOR  SPECIAL  PURPOSES  179 

terminals,  with  a  constant  voltage  across  the  primary 
terminals.  The  variations  in  the  secondary  voltage  are 
automatic,  being  the  result  of  very  small  changes  in  the 
secondary  current,  brought  about  by  switching  lamps  in 
or  out  of  the  circuit.  In  other  words,  the  secondary 
voltage  must  follow  as  nearly  as  possible  the  variations 
in  the  impedance  of  the  external  circuit,  so  that  a  doubled 
impedance  would  very  nearly  bring  about  a  doubhng  of 
the  secondary  voltage,  the  drop  in  current  being  as  small 
as  possible. 

Automatic  regulation  of  this  kind  may  be  obtained  by 
means  of  an  ordinary  transformer  having  a  large  amount 
of  magnetic  leakage,  as  for  instance  a  core  type  trans- 
former purposely  constructed  with  the  primary  turns 
on  one  limb  and  the  secondary  turns  on  the  other  limb, 
as  shown  diagrammatically  in  Fig.  i  of  Art.  2.  The 
vector  diagram  of  such  a  transformer  has  been  drawn  in 
Fig.  56,  based  on  the  simpHfied  diagram,  Fig.  48  (Art. 
36),  which  should  be  consulted  for  the  meaning  of  the 
vectors.  The  same  notation  has  been  used  in  Fig.  56 
as  in  Fig.  48,  and  it  is  to  be  observed  that,  on  account  of 
the  leakage  flux  being  a  large  percentage  of  the  useful 
flux,  a  small  reduction  in  the  current,  from  /i  to  I'l,  will 
automatically  cause  the  vector  Ee  (which  is  a  measure  of 
the  secondary  voltage)  to  become  £/,  just  twice  as  great. 

Although  by  suitably  designing  a  transformer  with 
considerable  leakage  flux,  a  small  reduction  in  the  reactive 
drop  (the  vector  hXp  of  Fig.  56)  wiH  produce  a  large 
increase  in  the  secondary  voltage,  it  is  obvious  that  still 
better  results  would  be  obtained  if  the  reactance  (or 
amount  of  leakage  flux)  could  be  made  to  decrease  at  a 


180  PRINCIPLES  OF  TRANSFORMER  DESIGN 

greater  rate  than  the  current.  Thus,  if  an  increase  of 
current  could  be  made  to  bring  about  a  change  in  the 
permeance  of  the  leakage  paths,  the  reactive  drop, 
instead  of  being  proportional  to  the  current,  might  be 
made  to  increase  at  a  greater  rate  than  the  current, 


Fig.  56.— Vector  Diagram  of  Transformer  with  Large  Amount  of  Leak- 
age Flux. 


and  so  bring  about  the  condition  illustrated  by  Fig,  57 
where  the  same  result, — i.e.,  a  doubling  of  the  secondary- 
voltage — is  seen  to  be  brought  about  by  a  very  much 
smaller  reduction  in  the  amount  of  the  current. 

It  is  evident   that   the  primary  volt-amperes  must 
remain  practically  constant  at  all  loads,  and  the  fact  that 


TRANSFORMERS  FOR  SPECIAL  PURPOSES  181 

the  actual  secondary  output  may  vary  considerably 
with  changes  in  the  resistance  of  the  external  circuit,  is 
accounted  for  by  the  alteration  in  the  power  factor  of 
the  primary  circuit.  Thus,  since  the  input  and  output 
of  a  transformer  must  be  the  same  except  for  the  internal 
losses,  the  changes  of  input  with  an  almost  constant 


>Ep 


Fig.    57.— Vector   Diagram    of   Transformer   with    Variable   Leakage 
Reactance. 


Epip  product  are  accounted  for  by  the  changes  in  the 
angle  4>  of  Fig.  57. 

Fig.  58  illustrates  the  principle  of  construction  of  the 
constant-current  transformer  with  variable  magnetic 
leakage.  One  coil  is  stationary  while  the  other  is  movable, 
being  suspended  from  a  pivoted  arm  provided  with  a 
counterweight,  and  free  to  slide  up  and  down  on  the  can- 


182 


PRINCIPLES  OF  TRANSFORMER  DESIGN 


tral  core  of  a  shell-type  magnetic  circuit.  The  movable 
coil  may  be  either  the  primary  or  the  secondary,  and  by 
careful  adjustment  of  the  balance  weight,  a  very  small 
change  in  the  current  may  be  made  to  produce  a  con- 
siderable change  in  the  relative  position  of  the  coils,  thus 
greatly  altering  the  relation  between  the  leakage  and 


Fig.  58. — Constant  Current  Transformer  with  "  Floating  "  Coih 


useful  flux  components,  the  (vectorial)  sum  of  which — 
passing  through  the  primary  coil — must  always  remain 
practically  constant. 

With  the  two  coils  in  contact,  the  maximum  secondary 
voltage — corresponding  to  the  maximum  number  of 
lamps  in  series — is  obtained;  while  on  short-circuit  the 
movable  coil  will  be  pushed  as  far  away  from  the  sta- 


TRANSFORMERS  FOR  SPECIAL  PURPOSES  183 

tionary  coil  as  the  construction  of  the  transformer  will 
admit.  Except  for  the  difficulty  of  calculating  accu- 
rately the  amount  of  the  leakage  flux-linkages  corre- 
sponding to  these  two  conditions,  the  design  of  a 
constant-current  transformer  for  any  given  output  is 
a  simple  matter.  Regulation  is  not  usually  required 
over  a  range  greater  than  from  full  load  to  about  one- 
third  of  full  load,  and  this  can  be  obtained  with  a  cur- 
rent variation  not  exceeding  i  per  cent. 

The  force  tending  to  move  the  coils  apart  can  readily 
be  calculated  with  the  aid  of  Formula  (4)  Art.  9;  but 
since  the  quantity  5am  cannot  be  predetermined  with 
great  accuracy — except  in  the  case  of  standard  designs 
for  which  data  have  been  accumulated — final  adjustments 
must  be  made  after  completion,  by  the  proper  setting  of 
the  counterweight. 

Constant-current  transformers  for  arc-lamp  circuits 
off  constant  pressure  mains  require  a  secondary  current 
between  6.5  and  10  amperes,  and  they  usually  operate 
in  conjunction  with  a  mercury  arc  rectifier  to  change  the 
alternating  current  into  a  continuous  current.  Trans- 
formers for  small  outputs  may  be  air  cooled,  while  the 
larger  units  should  be  oil-immersed  and,  if  necessary, 
cooled  by  circulating  water. 

The  full-load  efficiency  of  constant-current  trans- 
formers with  movable  coils  for  use  on  2200-volt  circuits 
ranges  from  90  per  cent  for  3  kw.  output  on  60-cycle 
circuits  to  96  per  cent  for  30  kw.  output  on  25-cycle 
circuits. 

47.  Current  Transformers  for  Use  with  Measuring 
Instruments.     These  transformers  are  of  comparatively 


184  PRINCIPLES  OF  TRANSFORMER  DESIGN 

small  size,  their  chief  function  being  to  provide  a  current 
for  measuring-instruments  which  shall  be  as  nearly  as 
possible  proportional  to  the  line  current  passing  through 
the  primary  coils.  By  their  use  it  is  possible  to  trans- 
form very  large  currents  to  a  current  of  a  few  amperes 
which  may  conveniently  be  carried  to  instruments  of 
standard  construction  mounted  on  the  switchboard 
panels  or  in  any  convenient  position  preferably  not  very 
far  removed  from  the  primary  circuit.  Again,  in  the 
case  of  high-potential  circuits,  even  if  the  reduction  of 
current  is  not  of  great  importance,  the  fact  that  the  sec- 
ondary circuit  of  the  current  transformer  can  be  at 
ground  potential  renders  unnecessary  the  special  instru- 
ments and  costly  methods  of  insulation  that  would  be 
required  if  the  Hne  current  of  high-voltage  systems  were 
taken  through  the  measuring  instruments. 

A  current  transformer  does  not  differ  fundamentally 
from  a  potential  transformer;  but  since  the  primary 
coils  are  in  series  with  the  primary  circuit,  the  voltage 
across  the  terminals  will  depend  upon  the  induced 
volts,  which,  in  their  turn,  depend  upon  the  impe- 
dance of  the  secondary  circuit.  With  the  secondary 
short-circuited,  the  voltage  absorbed  will  be  a  mini- 
mum, and  the  input  of  the  transformer  will  be  approxi- 
mately equal  to  the  copper  losses,  because  a  very  small 
amount  of  flux  will  then  be  sufficient  to  generate  the 
required  voltage,  and  the  iron  losses  will  be  neghgible. 

The  vector  diagram  for  a  series  transformer  does  not 
differ  from  that  of  a  shunt  transformer,  but  Figs.  59 
and  60  have  been  drawn  to  show  clearly  the  influence 
of  the  magnetizing  current  on   the  relation  between 


TRANSFORMERS  FOR  SPECIAL  PURPOSES 


185 


the  total  primary  and  secondary  currents.  Fig.  59 
shows  the  vector  relations  when  the  power  factor  is 
unity,  while  in  Fig.  60  there  is  an  appreciable  lag  be- 
tween the  current  and  e.m.f.  in  the  secondary  circuit. 

When  a  current  transformer  is  used  in  connection 
with  an  ammeter  only,   the  essential  condition  to  be 

fulfilled    is    that    the    ratio    7-1  or    7-)  be   as  nearly 

■lp\         ^  pI 

constant  as  possible  over  the  whole  range  of  current 
values.    When  the  secondary  current  is  passed  through 


Fig.  59. 


the  series  coil  of  a  wattmeter,  it  Is  equally  important 
that  Is  be  as  nearly  as  possible  opposite  in  phase  to 
/p,  or,  in  other  words,  that  the  angle  IpOI\  be  very  small. 

A  diagram,  such  as  Fig.  60,  may  be  constructed  for 
any  given  condition  of  load,  the  amount  of  the  flux 
B — and  therefore  the  exciting  current  /« — being  de- 
pendent upon  the  impedance  of  the  secondary  circuit, 
since  this  determines  the  necessary  secondary  voltage. 

On  the  sine  wave  assumption,  it  is  an  easy  matter 
to  express  the  quantity  Oly  in  terms  of  the  secondary 


186  PRINCIPLES  OF  TRANSFORMER  DESIGN 

current  and  the  two  components  of  the  exciting  current. 
The  vector  01 1  is  a  measure  of  the  secondary  current, 

being  simply  ^Ajt),  and  it  is  easily  seen  that 

Ip  =  Vlh  sin  9+/o)24-(/i  cos  0+/«;)2, 

whence  the  ratio  -^  can  be  calculated  for  any  power 
Ip 

factor  (cos  6)  and  any  values  of  the  secondary  current 


Fig.  6o. 


and  voltage.  It  is  interesting  to  note  that,  on  a  load 
of  unity  power  factor  (cos  d  =  i),  the  magnetizing  com- 
ponent of  the  total  exciting  current  does  not  appre- 
ciably affect  the  relation  between  the  magnitudes  of 
the  primary  and  secondary  currents,  and  for  all  practical 


TRANSFORMERS  FOR  SPECIAL  PURPOSES  187 

purposes  the  difference,  under  this  particular  load  con- 
dition, is 

_       _       ,  iron  loss  (watts) 

Ip  —  Il=Iw  = 


e.m.f .  induced  in  primary  (volts) 

If  this  difference  were  always  proportional  to  the 
primary  current,  there  would  be  no  particular  advantage 
in  keeping  it  very  small;  but  since  the  power  factor 
is  not  always  unity,  and  variations  in  current  mag- 
nitudes may  be  brought  about  by  phase  differences, 
it  is  always  advisable  to  aim  at  obtaining  an  exciting 
current  which  shall  be  a  very  small  percentage  of  the 
total  primary  current. 

The  phase  difference  between  Ip  and  h  (see  Figs. 
59  and  60)  may  be  expressed  as 

Angle  IpOh  =  tan-i  j -—y 

This  angle  must  be  very  small,  especially  when  the 
transformer  is  for  use  with  a  wattmeter.  It  should 
never  exceed  i  minute,  and  should  preferably  be  within 
thirty  seconds.  This  condition  can  only  be  satisfied, 
with  varying  values  of  6,  by  making  the  exciting  current 
(especially  the  magnetizing  component  /o)  very  small 
relatively  to  the  main  current.  It  is  therefore  neces- 
sary to  use  low  flux  densities  in  the  cores  of  series  trans- 
formers for  use  with  instruments,  and  this  incidentally 
leads  to  small  core  losses  and  a  small  "  energy  "  com- 
ponent (Iw)  of  the  total  exciting  current.  Flux  densities 
ranging  from  1500  to  2500  gausses  at  full  load  are  not 


188  PRINCIPLES  OF  TRANSFORMER  DESIGN 

uncommon  in  well-designed  series  instrument  trans- 
formers. Fig.  6 1  gives  approximate  losses  per  pound 
of  transformer  iron  for  these  low  densities  which  are 
not  included  in  the  curves  of  Fig.  27.  Although  curves 
for  alloyed  steel  are  not  given,  the  losses  may  be 
approximately  estimated  by  referring  to  Fig.  27  (Art. 
20)  and  noting  the  relative  positions  of  the  curves  for 
the  two  qualities  of  material. 

When  the  primary  current  is  large,  a  convenient 
form  of  current  transformer  is  one  with  a  single  turn  of 
primary,  that  is  to  say,  a  straight  bar  or  cable  passing 
through  the  opening  in  the  iron  core.  This  is  quite 
satisfactory  for  currents  of  1000  amperes  and  upward, 
and  the  construction  is  permissible  with  currents  as 
low  as  300  amperes,  especially  when  the  transformer 
is  to  be  used  in  cormection  with  a  single  ammeter,  i.e., 
without  a  wattmeter,  or  second  instrument,  or  relay 
coil,  in  series.  The  designer  should,  however,  aim  to 
get  1000  to  1500  ampere  turns,  or  more,  in  each  winding 
of  a  series  instrument  transformer. 

Although  the  presence  of  the  exciting  current  com- 
ponent of  an  iron-cored  transformer  renders  a  constant 
ratio  of  current  transformation  theoretically  unattain- 
able over  the  whole  range  of  current  values,  this  does 
not  mean  that  any  desired  ratio  of  transformation 
cannot  be  obtained  for  a  particular  value  of  the  primary 
current.  It  is,  of  course,  a  simple  matter  to  eliminate 
the  error  due  to  the  presence  of  the  exciting  current 

by  so  modifying    the    ratio  of  turns   (y^j    that    any 

desired  current  transformation  may  be  obtained  for  a 


TRANSFORMERS  FOR  SPECIAL  PURPOSES  189 


5000 


a  2000 


1000 


y 

"^ 

/ 

/ 

/ 

/ 

/ 

/■ 

/ 

1 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

f/ 

/ 

/ 

/ 

;/ 

/^ 

/ 

/ 

( 

^<:: 

y 

.^^ 

/ 

7 

A 

■/ 

/ 

/ 

/ 

/ 

/ 

( 

/ 

/ 

/ 

/ 

/ 

/ 

/ , 

/ 

/ 

/ 

/ 

/ 

/ 

V 

h 

/ 

/ 

V 

// 

/ 

/ 

0 

0 

1 

0 

2 

0 

3 

0 

4 

0 

5 

0 

6 

0 

7 

0 

8 

Total  watts  per  pound  =ti> 
Fig.  6i. — Losses  in  Transformer  Iron  at  Low  Flux  Densities. 


190 


PRINCIPLES  OF  TRANSFORMER  DESIGN 


specified  value  of  the  primary  current.  If  the  ratio  of 
transformation  is  correct  at  full  load,  it  will  be  prac- 
tically correct  over  the  range  from  f  to  full-load  cur- 
rent, the  error  being  most  noticeable  with  the  smaller 
values  of  the  main  current.  The  following  figures  are 
typical  of  the  manner  in  which  the  transformation 
ratio  of  series  instrument  transformers  is  likely  to  vary. 


Percentage  of  Full-load 
Current. 


lOO. 

75  • 
SO. 
25 

lO, 


Percentage  Departure  from  Full-load 
Ratio. 


30 
6.0 


0.04 
0.16 
05 
1.0 


o.S 

2.0 

6.0 

12.0 


Column  A  gives  average  values:  column  B  shows 
how  small  the  error  may  be  in  well-designed  transformers 
for  use  with  wattmeters  or  other  instruments  demand- 
ing constancy  in  the  current  ratios;  while  C  refers  to 
commercial  current  transformers  for  use  with  relays, 
trip  coils  of  switches,  and  other  apparatus  which  does 
not  call  for  great  accuracy  in  the  transforming  ratio. 
In  all  cases  a  fairly  low  power  factor  is  assumed,  and  a 
rated  full-load  output  of  about  50  volt-amperes.  If 
the  same  transformers  were  to  operate  on  an  external 
circuit  of  reduced  resistance  and  unity  power  factor, 
the  percentage  error  would  be  considerably  smaller. 

No  special  features  other  than  reliability  of  insulation, 
and  freedom  from  overheating  have  to  be  considered 


TRANSFORMERS  FOR  SPECIAL  PURPOSES  191 

in  connection  with  series  transformers  used  for  oper- 
ating regulating  devices  or  protective  apparatus  such 
as  trip  coils  on  automatic  overload  circuit-breakers. 
The  flux  density  in  the  core  may  then  be  higher  than 
in  instrument  transformers. 

48.  Auto-transformers.    An  ordinary  transformer  be- 
comes an  auto-transformer,  or  compensator,  when  the 


h 

Supply 

Voltage 

a 

\l\l\l\l\l\l\l\l\l 

I^'MM/'N'- 

0 

L_ 

'-'l 

'- 

'    Ic 
road  Vottag© 

i 

/; 


Fig.  62.— Ordinary  Transformer  Connected  as  Auto-transformer. 


connections  are  made  as  in  Fig.  62.  One  terminal  is 
then  common  to  both  circuits,  the  supply  voltage 
being  across  all  the  turns  of  both  windmgs  in  series, 
while  the  secondary  or  load  voltage  is  taken  off  a  por- 
tion only  of  the  total  number  of  turns.  This  arrange- 
ment would  be  adopted  for  stepping  down  the  voltage; 
but  by  interchanging  the  connections  from  the  supply 


192  PRINCIPLES  OF  TRANSFORMER  DESIGN 

circuit  and  the  load,  the  auto-transformer  can  be  used 
equally  well  for  stepping  up  the  voltage. 

There  is  little  advantage  to  be  gained  by  using  auto- 
transformers  when  the  ratio  of  transformation  is  large; 
but  for  small  percentage  differences  between  the  supply 
and  load  voltages,  considerable  economy  is  efifected  by 
using  an  auto-transformer  in  place  of  the  usual  type 
with  two  distinct  windings. 

Let  Tp  =  th.e  number  of  turns  between  terminals 

a  and  c  (Fig.  62) ; 
r,  =  the  number  of  turns  between  terminals 
c  and  b; 
then  {Tp-\-Ts)  =  the  number  of  turns  between  terminals 
a  and  b. 

The  meaning  of  other  symbols  is  indicated  on  Fig.  62. 
The  ratio  of  transformation  is 

'£     ~^ —    ^ ^54; 

If  used  as  an  ordinary  transformer,  the  transforming 
ratio  would  be 


jr  =  ^-l (55) 


The  ratio  of  currents  is 


M-' (5^) 


TRANSFORMERS  FOR  SPECIAL  PURPOSES  193 

while  the  current  h  in  the  portion  of  the  winding  com- 
mon to  both  primary  and  secondary  is  obtained  from 
the  equation 

J-c-L  s^^  J-  pi  Pi 

whence 

Ic  =  h{r-^), (57) 

or,  in  terms  of  the  secondary  current, 

i'=i{-^) (58) 

None  of  the  above  expressions  takes  account  of  the 
exciting  current  and  internal  losses. 

The  volt-ampere  output,  as  an  auto-transformer,  is 
Esis;  but  part  of  the  energy  passes  directly  from  the 
primary  into  the  secondary  circuit.  For  the  purpose 
of  determining  the  size  of  an  auto-transformer,  we 
require  to  know  its  equivalent  transformer  rating. 
The  volt-amperes  actually  transformed  are  EsL,  whence 

Output  as  ordinary  transformer _Ic_r  —  i      ,    . 
Output  as  auto-transformer        I3       r    ' 

which  shows  clearly  that  it  is  only  when  the  ratio  of 
voltage  transformation  {r)  is  small  that  an  appreciable 
saving  in  cost  can  be  effected  by  using  an  auto-trans- 
former. 

The  ratio  of  turns,  and  the  amount  of  the  currents 
to  be  carried  by  the  two  portions  of  the  winding  having 
been  determined  by  means  of  the  preceding  formulas, 
the  design  may  be  carried  out  exactly  as  for  an  ordi- 


194 


PRINCIPLES  OF  TRANSFORMER  DESIGN 


nary  potential  transformer,  attention  being  paid  to 
the  voltage  to  ground,  which  may  not  be  the  same  in 
the  auto-transformer  as  in  an  ordinary  transformer  for 
use  under  the  same  conditions.  Auto-transformers  are, 
however,  rarely  used  on  high  voltage  circuits,  although 
there  appears  to  be  no  objection  to  their  use  on  grounded 
systems. 

Efect  of  the  Exciting  Current  in  Auto-transformers. 
In  the  foregoing  discussions,  the  effect  of  the  exciting 
current  was  considered  neghgible.     This  assumption  is 


Supply 
— Er,— 


Tp  tarna 


OTMiTffoooooinJM^ooooooTnr 


V' 


Load 


Fig.  63. — Diagram  of  Connections  of  Auto-transformer. 


usually  permissible  in  practice;  but  since  it  may  some- 
times be  necessary  to  investigate  the  effect  of  the 
exciting  current  components,  a  means  of  drawing  the 
vector  diagram  showing  the  correct  relation  of  the 
current  components  will  now  be  explained. 

Fig.  63  is  similar  to  Fig.  62  except  that  it  shows 
the  connections  in  a  simplified  manner.  The  arrows 
indicate  what  we  shall  consider  the  positive  directions 
of  the  various  currents. 


TRANSFORMERS  FOR  SPECIAL  PURPOSES  195 

The  fundamental  condition  to  be  satisfied  is  that  the 
(vectorial)  addition  of  all  currents  flowing  to  or  from 
the  junction  c  or  b  shall  be  zero.     Whence, 

h-\-Is  =  Ic (60) 

Let  le  stand  for  the  exciting  current  when  there  is 
no  current  flowing  in  the  secondary  circuit.  This  is 
readily  calculated  exactly  as  for  an  ordinary  trans- 
former with  Ep  volts  across  {Tp-\-Ts)  turns  of  winding. 
Then,  since  the  resultant  exciting  ampere  turns  must 
always  be  approximately  {Tp-\-Ts)Ie,  the  condition  to 
be  satisfied  under  load  is 

IpTp-\-IcTs  =  Ie{Tp+Ts),      .     .     .     (61) 

which,  if  we  divide  by  Ts,  becomes 

{r-i)lp+lc  =  rle (62) 

If  le  in  this  equation  is  replaced  by  its  equivalent 
value  in  terms  of  the  other  current  components,  as 
given  by  Equation  (60),  we  get 

rh  =  rIe-L (63) 

The  vector  diagram  Fig.  64  satisfies  these  conditions; 
the  construction  being  as  follows: 

Draw  OB  and  OEs  to  represent  respectively  the  phase 
of  the  magnetic  flux  and  induced  voltage.  Draw  OL 
to  represent  the  current  in  the  secondary  circuit  in 
its  proper  phase  relation  to  Es.    Now  calculate  the 


196 


PRINCIPLES  OF  TRANSFORMER  DESIGN 


exciting  current  le  on  the  assumption  that  it  flows 

through  all  the  turns  {Tp-\-Ts),  and  draw  OM,  equal 

to  ric,  in  its  proper  phase  relation  to  OB.    Join  ML 

ML 

and  determine  the  point  C  by  making  Ch  = .     Then, 

r 

since  LM  is  the  vectorial  difference  between  rie  and 


Fig.   64. — Vector  Diagram  of  Auto-transformer,  Taking  Account  of 
Exciting  Current. 


rip,  whence  CIs=—Ii,,  and  CM  =  {r—i)Ip.  Also,  since 
OC  is  the  vectorial  sum  of  h  and  Ip,  it  follows  from 
Equation  (60)  that  OC  is  the  vector  of  the  current  Ic 
in  the  portion  of  the  winding  common  to  both  circuits. 
In  this  manner  the  correct  value  and  phase  relations 
of  the  currents  Ip  and  L,  in  the  sections  ac  and  cb  of 
the  winding,  can  be  calculated  for  any  given  load 
conditions. 


TRANSFORMERS  FOR  SPECIAL  PURPOSES  197 

49.  Induction  Regulators.  In  order  to  obtain  a  vari- 
able ratio  of  voltage  transformation,  it  is  necessary  either 
to  alter  the  ratio  of  turns  by  cutting  in  or  out  sections 
of  one  of  the  windings,  or  to  alter  the  effective  flux- 
linkages  by  causing  more  or  less  of  the  total  flux  Hnking 
with  the  primary  to  link  with  the  secondary. 

The  principle  of  variable  ratio  transformers  of  the 
moving  iron  type  is  illustrated  by  the  section  shown 
in  Fig.  65.  This  is  a  diagrammatic  representation  of 
a  single-phase  induction  regulator  with  the  primary 
coils  on  a  cyKndrical  iron  core  capable  of  rotation 
through  an  angle  of  90  degrees.  The  secondary  coils 
are  in  slots  in  the  stationary  portion  of  the  iron  cir- 
cuit. The  dotted  lines  show  the  general  direction  of 
the  magnetic  flux  when  the  primary  is  in  the  position 
corresponding  to  maximum  secondary  voltage.  As  the 
movable  core  is  rotated  either  to  the  right  or  left,  the 
secondary  voltage  will  decrease  until,  when  the  axis 
AB  occupies  the  position  CD,  the  flux  hnes  hnking 
with  the  secondary  generate  equal  but  opposite  e.m.f.s 
in  symmetrically  placed  secondary  coils,  with  the  result 
that  the  secondary  terminal  voltage  falls  to  zero.  If 
current  is  flowing  through  the  secondary  winding — 
as  will  be  the  case  when  the  transformer  is  connected 
up  as  a  "  booster  "  or  feeder  regulator — the  reactive 
voltage  due  to  flux  Hnes  set  up  by  the  secondary  current 
and  passing  through  the  movable  core  in  the  general 
direction  CD,  wiH  be  considerable  unless  a  short-circuited 
winding  of  about  the  same  cross-section  as  these  cond- 
ary  is  provided  as  indicated  in  Fig.  65. 

It  is  immaterial  whether  the  winding  on  the  movable 


198 


PRINCIPLES  OF  TRANSFORMER  DESIGN 


core  be  the  primary  or  secondary;  but  if  the  primary  is 
on  the  stationary  ring,  the  short-circuited  coils  must 
also  be  on  the  ring. 

The  chief  difficulty  in  the  design  of  induction  regulators 


Fig.  65.— Diagram  of  Single-phase  Variable-ratio  Transformer  of  the 
Moving-iron  Type. 


arises  from  the  introduction  of  necessary  clearance 
gaps  in  the  magnetic  circuit,  and  the  impossibility  of 
arranging  the  coils  as  satisfactorily  as  in  an  ordinary 
static  transformer  so  as  to  avoid   excessive  magnetic 


TRANSFORMERS  FOR  SPECIAL  PURPOSES  199 

leakage.  A  large  exciting  current  component  and  an 
appreciable  reactive  voltage  drop  are  characteristic  of 
the  induction  voltage-regulator. 

Fig.  66  is  a  diagram  showing  a  single-phase  regu- 
lating transformer  of  the  type  illustrated  in  Fig.  65 
connected  as  a  feeder  regulator,  the  secondary  being 
in  series  with  one  of  the  cables  leaving  a  generating  station 
to  supply  an  outlying  district.  The  movement  of  the 
iron  core  can  be  accomphshed  either  by  hand,  or  auto- 
matically by  means  of  a  small  motor  which  is  made  to 
rotate  in  either  direction  through  a  simple  device 
actuated  by  potential  coils  or  relays. 

The  lower  diagram  of  Fig.  66  shows  the  core  carrying 
the  primary  winding  in  the  position  which  brings  the 
voltage  generated  in  the  ring  winding  to  zero.  The 
flux  lines  shown  in  the  diagram  are  those  produced 
by  the  magnetizing  current  in  the  primary  winding; 
but  there  are  other  flux  lines — not  shown  in  the  diagram 
— which  are  due  to  the  current  in  the  ring  winding. 
It  is  true  that  the  movable  core  carries  a  short-cir- 
cuited winding — not  shown  in  Fig.  66 — which  greatly 
reduces  the  amount  of  this  secondary  leakage  flux; 
but  it  will  nevertheless  be  considerable,  and  the  secondary 
reactive  voltage  drop  is  likely  to  be  excessive,  especially 
if  the  ring  winding  consists  of  a  large  number  of  turns. 
An  improvement  suggested  by  the  writer  at  the  time  * 
when  this  type  of  apparatus  was  in  the  early  stages  of 
its  development,  consists  in  putting  approximately  half 
the  secondary  winding  on  the  portion  of  the  magnetic 
circuit  which  carries  the  primary  winding,  the  balance 

*The  year  1895. 


200  PRINCIPLES  OF  TRANSFORMER  DESIGN 


Fig    66- — Variable-ratio  Transformer  Connected  as  Feeder  Regulator. 


TRANSFORMERS  FOR  SPECIAL  PURPOSES  201 

of  the  secondary  turns  being  put  on  the  other  portion 
of  the  magnetic  circuit.  The  connections  are  made  as  in 
Fig.  67,  the  result  being  that  the  movement  of  the 
rotating  core,  to  produce  the  full  range  of  secondary- 
voltage  from  zero  to  the  desired  maximum,  is  now  180° 
instead  of  90°  as  in  Fig.  66;  but  since,  under  the  same 
conditions  of  operation,  the  ring  winding  for  a  given 
section  of  iron  will  carry  only  half  the  number  of  turns 
that  would  be  necessary  with  the  ordinary  type  (Fig. 
66),  the  secondary  reactive  voltage  drop  is  very  nearly 
halved.  This  is  one  of  the  special  features  of  the 
regulating  transformers  manufactured  by  Messrs. 
Switchgear  &  Cowans,  Ltd.,  of  Manchester,  England. 
Consider  the  case  of  a  single-phase  system  with  2200 
volts  on  the  bus  bars  in  the  generating  station.  The 
voltage  drop  in  a  long  outgoing  feeder  may  be  such  as 
to  require  the  addition  of  200  volts  at  full  load  in  order 
to  maintain  the  proper  pressure  at  the  distant  end.  If 
this  feeder  carries  100  amperes  at  full  load,  the  neces- 
sary capacity  of  a  boosting  transformer  of  the  type 
shown  diagrammatically  in  Fig.  67  is  20  k.v.a.  This 
variable-ratio  transformer,  with  its  primary  across  the 
2 200- volt  supply,  and  its  secondary  in  series  with  the 
outgoing  feeder,  will  be  capable  of  adding  any  voltage 
between  o  and  200  to  the  bus-bar  voltage.  As  an 
alternative,  the  supply  voltage  at  the  generating  station 
end  of  this  feeder  may  be  permanently  raised  to  2300 
volts  by  providing  a  fixed-ratio  static  transformer 
external  to  the  variable-ratio  induction  regulator  and 
connected  with  its  secondary  in  series  with  the  feeder. 
An  induction  regulator  of  the  ordinary  type  (Fig.  66) 


202  PRINCIPLES  OF  TRANSFORMER  DESIGN 


Fig.  67.— Moving-iron  Type  of  Feeder  Regulatorwith  Specially Drranged 
Secondary  Winding. 


TRANSFORMERS  FOR  SPECIAL  PURPOSES  203 

capable  of  both  increasing  and  decreasing  the  pressure 
by  lOO  volts,  will  then  provide  the  desired  regulation 
between  2200  and  2400  volts.     The  equivalent  trans- 

r  r      ,  .  ,  Ml    1        looXioo 

former  output  of  this  regulator  will  be  =  10 

1000 

k.v.a. 

The  Polyphase  Induction  Regulator.  Two  or  three 
single-phase  regulators  of  the  type  illustrated  in  Fig.  65 
may  be  used  for  the  regulation  of  three-phase  circuits; 
but  a  three-phase  regulator  is  generally  preferable.  The 
three-phase  regulator  of  the  inductor  type  is  essentially 
a  polyphase  motor  with  coil-wound — not  squirrel-cage 
— rotor,  which  is  not  free  to  rotate,  but  can  be  moved 
through  the  required  angle  by  mechanical  gearing  oper- 
ated in  the  same  manner  as  the  single-phase  regulator. 
The  rotating  j&eld  due  to  the  currents  in  the  stator  coils 
induces  in  the  rotor  coils  e.m.f.'s  of  which  the  magnitude 
is  constant,  since  it  depends  upon  the  ratio  of  turns, 
but  of  which  the  phase  relation  to  the  prhnary  e.m.f. 
depends  upon  the  position  of  the  rotor  coils  relatively 
to  the  stator  coils.  When  connected  as  a  voltage 
regulator  for  a  three-phase  feeder,  the  vectorial  sum  of 
the  secondary  and  primary  volts  of  a  three-phase 
induction  regulator  will  depend  upon  the  angular  dis- 
placement of  the  secondary  coils  relatively  to  the  cor- 
responding primary  coils. 

Mr.  G.  H.  Eardley-Wihnot  *  has  pointed  out  certain 

advantages  resulting  from  the  use  of  two  three-phase 

induction  regulators  with  secondaries  connected  in  series, 

for  the  regulation  of  a  three-phase  feeder.    By  making 

*  The  Electrician,  Feb.  19,  1915,  Vol.  74,  page  660. 


204  PRINCIPLES  OF  TRANSFORMER  DESIGN 

the  connections  so  that  the  magnetic  fields  in  the  two 
regulators  rotate  in  opposite  directions,  the  resultant 
secondary  voltage  will  be  in  phase  with  the  primary 
voltage.  The  torque  of  one  regulator  can  be  made  to 
balance  that  of  the  other,  thus  greatly  reducing  the 
power  necessary  to  operate  the  controlling  mechanism. 


INDEX 


A 

PAGE 

Absolute  unit  of  current 26 

Air-blast,  cooling  by, .' 88 

All-day  efficiency  (see  Efficiency). 

Alloyed-iron  transformer  stampings 19 

Ampere-turns  to  overcome  reluctance  of  joints 127 

Analogy  between  dielectric,  and  magnetic,  circuits $^ 

Auto-transformers 191 


B 

B-H  curves  (see  Magnetization  curves). 

Bracing  transformer  coils  {see  Stresses  in  transformer  coils). 

Bushings  {sec  Terminals). 

C 

Calorie,  definition 99 

Capacity  current 41 

electrostatic 33, 36 

of  plate  condenser 40 

Capacities  in  series 42 

Charging  current   (Capacity  current) 41 

Classification  of  transformers 14 

Compensators 191 

Condensers  in  series 42 

Condenser  type  of  bushing 62 

Conductivity,  heat 80,  82,  87 

Constant-current  transformers 178 

Construction  of  transformers 17,  24,  31 

205 


206  INDEX 

PAGE 

Cooling  of  transformers 14,  88,  gi,  103 

by  air  blast 88 

forced  oil  circulation 106 

water  circulation 105 

Copper  losses 72,  75,  76,  83,  142,  165 

resistivity  of 144 

space  factor  (see  Winding  space  factor). 

Core  loss  (usual  values)  (see  also  Losses  in  iron) 77 

Core-type  transformers 17,  22 

Corrugations,  effect  of,  on  sides  of  tank 94 

on  insulator  surface 60 

Coulomb 34 

Current  density  in  windings 72 

transformers 184 


D 

Density  (see  Flux-  and  Current-density). 

Design  coefficient  (c) 149 

numerical  example  in 154 

problems 13 

procedure  in 150 

Dielectric  circuit 32 

constant 36 

constants,  table  of 37 

strengths,  table  of 37 

Disruptive  gradient 36,  62 

Distributing  transformers 17 


E 

Eddy  currents  in  copper  windings 73 

current  losses  (see  Losses). 

Efifective  cooling  surface  of  tanks :     96 

Efficiency 73,  167 

all-day 74 

approximate,  of  commercial  transformers 74,  183 

calculation  of,  for  any  power  factor 77 

maximum 145 

Elastance,  definition 35 


INDEX  207 

PAGE 

Electrifying  force 38 

Electrostatic  force 38 

E.m.f.  in  transformer  coils  {see  also  Volts;  Voltage) 4,  5,  6 

Equivalent  cooling  surface  of  tanks 96 

ohmic  voltage  drop 134,  137 

Exciting  current S,  125,  168 

in  auto-transformers 194 

volt-amperes 129 

(curves) 131 

F 

Farad 33 

Flux  density,  electrostatic 35 

in  transformer  cores 72,  164 

leakage  {see  Leakage  flux). 

Forces  acting  on  transformer  coils 24,  1 74 

Frequency,  effect  of,  on  choice  of  iron 19 

allowance  for  core  loss 77 

Furnaces,  transformers  for  electric 177 

H 

Heat  conductivity  of  materials 80 

copper 83,  87 

insulation 87 

Heating  of  transformers  {see  Temperature  rise). 

High-voltage  testing  transformers 15 

Hottest  spot  calculations 84 

Hysteresis,  losses  due  to  {see  Losses). 

I 

Induction  regulator 197 

polyphase 203 

Instrument  transformers 183 

Insulation  of  end  turns  of  transformer  windings 50 

oil 52 

problems  of  transformer ; 32 

thickness  of 48 

Iron,  losses  in 69,  77,  142,  i8g 


208  INDEX 


L 

PAGE 

Laminations,  losses  in 69,  77,  142,  189 

shape  of,  in  shell-type  transformer 19 

thickness  of 19 

Large  transformers 16,  17 

Leakage  flux 98,  107,  1 18,  1 79,  198 

reactance  {see  Reactance;  Reacti\'e  voltage  drop). 

Losses,  eddy  current 69 

hysteresis 69 

in  copper  windings 72,  75,  76,  83,  142,  165 

in  iron  circuit 69,  77,  142,  189 

power,  in  transformers 69 

ratio  of  copper  to  iron 145 

M 

Magnetic  leakage  {sec  Leakage  flux). 

Magnetization  curves  for  transformer  iron 128 

Magnetizing  current  {see  Exciting  current). 

Mechanical  stresses  in  transformers 24,  1 74 

Microfarad 36 


O 

Oil  insulation 52 

Output  equation ' 138 

Overloads,  effect  of,  on  temperature 98 

P 

Permeanc^ 34,  39 

Permittance  {sec  Capacity). 

Polyphase  transformers 12,22 

Potential  gradient 38 

Power  losses  {sec  Losses). 

transformers 16,  154 

Q 

Quantity  of  electricity  (Coulomb) 34 


INDEX  209 


R 

PAGE 

Reactance,  leakage,  experimental  determination  of 114 

Reactive  voltage  drop 117,  137,  180 

Regulation 109,  132,  168 

formulas 134,  135 

Regulating  tranformers 197 

polyphase 203 

Reluctance,  magnetic 35 

Resistance  of  windings 165 

thermal 81 

Resistivity  of  copper 144 


S 

"  Sandwiched  "  coils 118 

Saturation;  reasons  for  avoiding  high  flux  densities 164 

Self-induction  of  secondary  winding 108 

Series  transformers 184 

Shell-type  transformers 17,  20,  24,  155 

Short-circuited  transformer,  diagram  of 116 

Silicon-steel  for  transformer  stampings 71 

Single-phase  units  used  for  three-phase  circuits 12 

Space  factor,  copper  {see  Winding  space  factor). 

iron 151 

Sparking  distance;  in  air 58,  68 

in  oil 52,  53 

Specifications 140,  155 

Specific  inductive  capacity  (see  Dielectric  constant). 

heat;  of  copper 99 

of  oil 99 

Stacking  factor 151 

Stampings,  transformer,  thickness  of 19 

Static  shield  on  h.t.  terminals 65,  68 

Stresses  in  transformer  coils 24,  1 74 

Surface  leakage 46 

under  oil 54 

Symbols,  list  of ix 


210  INDEX 


T 

PAGE 

Temperature  rise  of  transformers 79,  90,  92,  94,  98,  170 

after  overload  of  short  duration 99 

Terminals 54 

composition-filled 59 

condenser  type 6a 

oil-filled 57,  60 

porcelain 57 

Test  voltages 58 

Theory  of  transformer,  elementary 2 

Thermal  conductivity  {see  Heat  conductivity). 

ohm,  definition 81 

Three-phase  transformers 12,22 

Transformers,  auto 191 

constant  current 178 

core-type 17,  20,  22 

current 184 

distributing 17 

for  electric  furnaces 177 

large  currents 178 

use  with  measuring  instruments 183 

polyphase 12,22 

power 16,  154 

series 184 

shell-type 17,  20,  24,  155 

welding 177 

Tubular  type  of  transformer  tank 104 


V 

Variable-ratio  transformers '. . . .   197 

Vector  diagram  illustrating  effect  of  leakage  flux no,  112 

of  auto-transformer 196 

short-circuited  transformer 116 

series  transformer 185,  186 

transformer  on  inductive  load 11,  133,  134,  135 

non-inductive  load 10 


INDEX  211 

PAGE 

Vector  diagram  of  transformer  with  large  amount  of  leakage  flux. . .  i8o 

open  secondary  circuit 5 

variable  leakage  reactance 181 

showing  components  of  exciting  current 126 

Voltage,  effect  of,  on  design 15 

drop  due  to  leakage  flux 117,  137 

regulation  {see  Regulation). 

Voits  per  turn  of  winding 141 


W 

Water-cooled  transfoniiers 105 

Weight  of  transformers 151,  173 

Welding  transformers 177 

Windings,  estimate  of  number  of  turns  in 141 

Winding  space  factor ■ 51,  151,  152 

'Window,"  dimensions  of,  in  shell-type  transformers 160,  163 

U'ire,  size  of,  in  windings  {see  Current  density). 


SUPPLEMENTARY   INDEX 

OF   TABLES,   CURVES,   AND   FORMULAS 
A 

PAGE 

Air  clearances  (Formula) 49 

quantity  required  for  air-blast  cooling 89,  90 

Ampere-turns,  allowance  for  joints 127 

B 

B-H  curves  (Gausses  and  amp-turns  per  inch) 128 

C 

Capacity  current 42 

in  terms  of  dimensions,  etc 36 

Charging  current 42 

Cooling  area  of  tanks  (Curve) 93 

Copper  space  factors 51,  151,  152 

Core  loss 70,  77,  189 

Corrugated  tanks,  correction  factor  for  cooling  surface  of 96 

Current  density  (usual  values) 72 

D 

Density,  current,  in  coils  (usual  ^'alues) 72 

in  transformer  cores  (Table) 72 

Dielectric  constants  (Table) 37 

strengths  (Table) 37 

Disruptive  gradient  {see  Dielectric  strength). 

E 

EflBciency  (usual  values) 74 

E.m.f .,  formulas 5,  6 

21R 


214  SUPPLEMENTARY   INDEX 

PAGE 

Equivalent  surface  of  corrugated  tanks  (correction  factor) g6 

Exciting  volt-amperes,  Formula 130 

Curve 131 


Flux  densities  in  core  (Table) 72 

Force  exerted  cii  coil  bj'  leakage  flux 28 

H 

Hottest  spot  tempc-ature  (Formula) 86 

I 

Inductive  voltage  drop  (Formula) 124 

Insulation,  air  clearance 49 

oil  clearance 53 

thickness  of  (Table) 48 

Iron  loss  (Curv-es) 70, 189 

J 

Joints  in  iron  circuit,  ampere  turns  required  for 127 


Losses  in  cores  (usual  values) 77 

transformer  iron  (Curves) 70,  189 


M 

Magnetization  curves  for  transformer  iron 128 

Magnetizing  volt-amperes  (Curve) 131 

Mechanical  force  on  coil  due  to  magnetic  field 28 


O 

Oil,  insulation  thickness  in S3,  54 

transformer,  test  voltages 52 

Output  equation 138 


SUPPLEMENTARY   INDEX  215 


P 

PAGH 

Power  losses  in  transformer  iron  (Curves) 70,  189 


R 

Reactance,  leakage,  in  terms  of  test  data 117 

Reactive  voltage  drop  (Formula) 124 

Regulation  formulas 135,  136 

Resistance,  equivalent  primary 117 


S 

Space  factors,  copper 51,  151,  152 

'  iron 151 

Specific  inductive  capacity  (Dielectric  constant),  (Table) 37 

Surface  leakage  distance,  in  air 50 


under  oil . 


54 


T 

Temperature  of  hottest  spot  (Formula) 86 

rise  due  to  overloads  (Formula) g8,  loi 

in  terms  of  tank  area  (Curve) 93 

Thickness  of  insulation 48 

iioil 53,54 

V 

Voltage  drop,  reactive  (Formula) 124 

regulation  (Formulas) 135,  136 

Volt-amperes  of  excitation,  (Formula) 130 

(Curves) 131 

Volts  per  turn  of  winding  (Formula) 142 


numerical  constants. 


W 


149 


Water,  amount  of,  required  for  water-cooling  coils , 105 

Winding  space  factors 51,  151,  152 


216  SUPPLEMENTARY   INDEX 


NUMERICAL  EX.'^MPLES 

PAGE 

Capacities  in  series , 43 

Composition-filled  bushing 58 

Condenser-type  bushing 65 

Cooling-coil  for  water-cooled  transformers 105,  171 

"Hottest  spot"  temperature  calculation 87 

Layers  of  different  insulation  in  series 44 

Mechanical  stresses  in  coils 174 

Plate  condenser 41 

Temperature  rise  due  to  overloads 98,  102 

of  self  cooling  oil-immersed  transformer 94 

with  tank  having  corrugated  sides 97 

Transformer  design 154 

Voltage  regulation 136,  168 

Volt-amperes  of  excitation  per  pound  of  iron  in  core 130 


MOPMTT  LIBRARY 

II.CStaU  CoOege 


